If you have finished with the activities on the unit try to these exercises.

Repasa y practica con estas actividades de la BBC y juega mientras practicas.

Decimals

Place value and ordering decimals

Decimal place values

We use a decimal point to separate units from parts of a whole (like tenths, hundredths, thousandths, etc).

0.1 is a tenth, 1/10 of a unit

0.01 is a hundredth, 1/100 of a unit

0.001 is a thousandth. 1/1000 of a unit

In 52.13, the value of the figure 1 is 1/10 , and the value of the figure 3 is 3/100.

Ordering decimals

When ordering numbers, always compare the left digits first.

Eg Which is greater, 2.301 or 2.32?

Units		Tenths	Hundredths	Thousandths
2	.	3	0	1
2	.	3	2

Both numbers have two units and three tenths, but 2.301 has no hundredths, whereas 2.32 has two hundredths. Therefore, 2.32 is greater than 2.301.

Adding a zero

Another way to look at it is to add a zero to the end of 2.32 (this doesn’t change its value as it’s after the decimal point).

The two numbers are now 2.320 and 2.301 and it is quite easy to see that 2.320 is bigger (just as 2 320 is bigger than 2 301).

Questions

Q1. In the number 3.546, what is the value of the figure 4?

Q2. Place the following numbers in order, smallest first: 3.2, 3.197, 3.02, 3.19

Adding and subtracting decimals

When adding and subtracting decimals, remember is to keep the decimal points in line in the question and the answer.

Adding decimals

Question

David is doing some DIY. He buys a 2m length of wood. He needs to cut two pieces of wood - one of length 0.6m and one of length 1.02m.

What is the total length of wood that David needs to cut?

Subtracting decimals

Question

David originally had 2m of wood. What is the length of the piece of wood that is left?

Multiplying decimals by 10, 100 and 1000

Multiplying by 10

When a decimal is multiplied by 10, every figure moves one place to the left.

What is 4.25 x 10? 42.5

Multiplying by 100

When multiplying by 100, every figure moves two places to the left.

Question

Which is bigger: 0.005 × 10 or 0.0004 × 1000?

Dividing decimals by 10, 100, 1000

Dividing by 10

When you divide by 10, every figure moves one place to the right. Hundreds become tens, tens become units, units become tenths and tenths become hundredths.

Dividing a decimal by 10

What is 27 divided by 10?

Dividing by 100

When you divide by 100, every figure moves two places to the right.

Dividing a decimal by 100

What is 27 divided by 100?

Dividing by 1000

When you divide by 1000, every figure moves three places to the right.

Dividing a decimal by 1000

What is 30 divided by 1000?

Multiplying a decimal by a whole number

Multiplying a decimal by a whole number is the same as multiplying two whole numbers. Remember:

If there is one digit after the decimal point in the question, there will be one digit after the decimal point in the answer.

If there are two digits after the decimal point in the question, there will be two digits after the decimal point in the answer.

Question

Calculate:

a) 2.43 × 7
b) 2.4 × 5
a) There were two digits after the decimal point in the question (4 and 3), so you must have two digits after the decimal point in the answer.

b) There was one digit after the decimal point in the question, so you must have one digit after the decimal point in the answer. The answer is therefore 12.0, but this can then be given as 12.

Check that you have a sensible answer by finding an approximate solution.

In the above example you were asked to calculate 2.4 × 5.

2 × 5 = 10, so you are looking for an answer which is slightly bigger than 10. So an answer of 12 seems sensible.

Dividing a decimal by a whole number

Remember to keep the decimal points aligned in the question and the answer.

Example

Work out 4.05 divided by 9

Solution:

Example

Work out 2.4 divided by 5

Solution:

It is sometimes necessary to add a ‘0′ or ‘0’s to the end of a decimal, as in this example (2.40 is the same as 2.4 but the question stays the same)

Multiplying by a number between 0 and 1

The multiplication sign can be replaced by ‘lots of’.

For example,

2 × 3 means 2 lots of 3
6 × 8 means 6 lots of 8

So, 1/2 × 10 means 1/2 of 10
And 1/3 × 12 means 1/3 of 12

When you multiply by a number greater than 1, you get an answer that is greater than the original number. But when you multiply by a number between 0 and 1, the answer is smaller than the original number.

In general:

m × 1/n = m ÷ n

Example

8 × 1/4 = 8 ÷ 4 = 2

20 × 1/5 = 20 ÷ 5 = 4

Dividing by a number between 0 and 1

Imagine that you had 10 bars of chocolate that you wanted to share amongst some children.

If you gave the children 2 bars each, you would have enough for 5 children.
10 ÷ 2 = 5

If you gave the children 1/2 bar each, you would have enough for 20 children.
10 ÷ 1/2 = 20

The pattern

Can you see what’s happening?

10 ÷ 2 = 5

10 ÷ 1/2 = 20

When you divide by a whole number the answer is less than the original number. When you divide by 1/2 the answer (20) is greater than the original number (10).

It’s the opposite of multiplying. When we divide by a number greater than 1, we get an answer that is less than the original number. But when we divide by a number between 0 and 1 the answer is larger than the original number.

So, 10 ÷ 1/2 = 20

Similarly, 10 ÷ 1/3 = 30 and 10 ÷ 1/4 = 40

In general:

m ÷ 1/n = mn

Questions

Q1. What is 10 ÷ 1/7 ?

Q2. Find the value of: 4 ÷ 1/3

BBC

TEST

Play games with decimals.

CONCEPT OF A FRACTION

If you cut a cake into two equal pieces and eat one of them, you have eaten ¹/₂ (half) a cake.

If a cake is cut into five equal pieces and you eat three of them, you have eaten ³/₅ (three fifths) of a cake.

¹/₂ and ³/₅ are examples of fractions - parts of a whole.

HOW TO READ FRACTIONS

To read a fraction in English you have to read the numerator and then the denominator with the ordinal. You can use plural if you have more than one part. For example, 1/3 is one third but 2/3 is two thirds. When the denominator is 2 you have to read “half”, 1/2 is one half, 5/2 is five halves. If the fractions is very big you can use “over”, 4/11 is four over eleven.

Equivalent fractions

Cutting the cake into six equal pieces and eating two is equivalent to cutting the cake into three equal pieces and eating one. You eat the same amount of cake in both cases.

Question

If the cake is cut into 12 equal pieces, how many will we have to eat in order to have the equivalent of 1/3 of the cake?

Common factors and simplest form

Common factors

The factors of a number are those numbers that divide into it exactly.

Numbers have common factors if the same number divides into both of them.

So 4 is a common factor of both 8 and 12, as it divides into both of them. 2 is a common factor of both 2 and 6, as it divides into both of them.

Simplest form

You know that ⁴/₁₂ = ²/₆ = ¹/₃

4 and 12 have a common factor (4), so ⁴/₁₂ can be written as ¹/₃ (divide the top and the bottom by 4).

2 and 6 have a common factor (2), so ²/₆ can be written as ¹/₃ (divide the top and the bottom by 2).

However, 1 and 3 have no common factors, so ¹/₃ cannot be simplified. When a fraction cannot be simplified we say that it is its simplest form.

Mixed numbers and improper fractions

A whole number can be written as ²/₂, ³/₃, ⁴/₄, etc.

So 1 ²/₃ can be written as

³/₃ + ²/₃ = ⁵/₃

Mixed numbers

1 ²/₃ is known as a mixed number, because it is made up of a whole number and a fraction.

Improper fractions

⁵/₃ is called an improper fraction, because the top number is bigger than the bottom number.

Converting from a mixed number to an improper fraction

You can write the whole number part as a fraction, then add the fractions together.

1 ²/₃ = ³/₃ + ²/₃ = ⁵/₃

Here is another example:

2 ¹/₄ = 1 + 1 + ¹/₄ = ⁴/₄ + ⁴/₄ + ¹/₄ = ⁹/₄

Converting from improper fractions to mixed numbers

You can separate out the fraction into smaller fractions, like this:

¹⁷/₅= ⁵/₅ + ⁵/₅ + ⁵/₅ + ²/₅ = 3 ²/₅

Another way to convert an improper fraction is to find how many whole numbers you get, by using a division.

For example let’s convert ¹⁷/₅ to a mixed number again.

We start by dividing the top number by the bottom number.
17 divided by 5 is 3 remainder 2.
So the whole number part is 3, and the remainder 2 means there are ²/₅ left over.

So the answer is ¹⁷/₅ = 3 ²/₅

Question

Write ²⁰/₇ as a mixed number.

Ordering fractions

Which fraction is bigger, ³/₄ or ⁵/₇ ?

It is hard to answer this question just by looking at the fractions. However, if you write the fractions with the same bottom number, the question will be easy.

³/₄ has a denominator of 4, and ⁵/₇ has a denominator of 7.

4 and 7 both divide into 28, so rewrite the fractions with a denominator of 28.

³/₄= ²¹/₂₈

⁵/₇= ²⁰/₂₈

It is easy to see that ²¹/₂₈ is bigger than ²⁰/₂₈.

Therefore ³/₄ is bigger than ⁵/₇.

To compare fractions, first write them with the same number at the bottom.

Adding and subtracting

It is hard to picture what the answer is if you add ¹/₂ and ¹/₃. Rewriting the fractions with a common bottom number (in this case, 6) makes it easier to see the answer.

Remember: You can only add and subtract fractions when the bottom numbers are the same.

So to add or subtract fractions:

1. Change the fractions so they have the same bottom number.
2. Add or subtract the top numbers.

Example

¹/₂ + ¹/₃ = ³/₆ + ²/₆ = ⁵/₆

⁷/₁₀ - ²/₅ = ⁷/₁₀ - ⁴/₁₀ = ³/₁₀

Question

What is ¹/₄ + ¹/₃ = ?

Multiplying fractions

¹/₂ of ¹/₂ = ¹/₂ × ¹/₂ = ¹/₄

²/₃ of ⁴/₅ = ²/₃ × ⁴/₅ = ⁸/₁₅

Multiply the top and bottom numbers then simplify where necessary.

Question

Calculate ³/₄ × ²/₅ = ?

Dividing fractions

When you divide 10 by 2, you are working out how many 2’s there are in 10.

10 ÷ 2 = 5, so there are five 2’s in 10.

In a similar way, when dividing 2 by ¹/₂, you are working out how many ¹/₂’s there are in 2.

There are four ¹/₂’s in 2, so 2 ÷ ¹/₂ = 4.

If you divide 1 ¹/₂ by ¹/₄ you are working out how many ¹/₄’s there are in 1 ¹/₂ .

There are six ¹/₄’s in 1 ¹/₂, so 1¹/₂ ÷ ¹/₄= 6.

Do you see a pattern? Let’s write out those calculations a different way.

• 2 ÷ ¹/₂ = 4 so 2 ÷ ¹/₂ is the same as 2 × 2
• 1¹/₂ ÷ ¹/₄ = ³/₂ ÷ ¹/₄ = 6
  so ³/₂ ÷ ¹/₄ is the same as ³/₂ × 4 = ¹²/₂ = 6

Remember: To divide fractions, turn the second fraction upside down, then multiply.

Question

Calculate ³/₄ ÷ ⁴/₅

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Now visit this site and do this exercises.

APUNTES DEL TEMA 2

1.- MULTIPLES AND FACTORS

1.1.- Concept of multiple.

1.2.- Concept of factor.

1.3.- The properties of multiples and factors.

 

2.- PRIME AND COMPOSITE NUMBERS

A prime number only has two factors: the number one and itself. For example: 3, 5, 11, 17, etc. A composite number has more than two factors. For example: 4, 9, 15, 30, etc.

3.- DIVISIBILITY RULES

Las reglas de divisibilidad te ayudan a saber si un número es múltiplo de otro sin hacer la división.

• Rule of number 2: A number is divisible by 2 if its last digit is either 0 or an even number. Un número es divisible por 2 si su última cifra es 0 ó un número par. Example: 46,200, 34, 108…..
• Rule of number 3: A number is divisible by 3 if the sum of its digits is a multiple of 3. Un número es divisible por 3 si la suma de sus cifras es múltiplo de 3. Example: 45, 105, 300, 417….
• Rule of number 4: A number is divisible by 4 if its two last digits are multiples of 4. Un número es divisible por 4 si sus dos últimas cifras son múltiplo de 4. Example: 100, 224, 340, 664….
• Rule of number 5: A number is divisible by 5 if it ends in 0 or 5. Un número es múltiplo de 5 si acaba en 0 ó 5. Example: 200, 345, 650, 800 …..
• Rule of number 9: A number is divisible by 9 if the sum of its digits is a multiple of 9. Un número es divisible por 9 si la suma de sus cifras es múltiplo de 9. Example: 81, 333, 450, 1278…..
• Rule of number 10: A number is divisible by 10 if it ends in 0. Un número es divisible por 10 si acaba en 0. Example: 30, 400, 500.
• Rule of number 11: A number is divisible by 11 if the difference between the sum of the digits on odd positions and the sum of the digits on even positions is 0, 11 or a multiple of 11. Un número es divisible por 11 si la diferencia entre la suma de las cifras en posición par y la suma de las cifras en posición impar es 0, 11 o un múltiplo de 11. Example: 121, 3652

Solve the following exercises:

• Use the divisibility rules to complete the following table:

Divisible by	2	3	4	5	9	10	11	25	100
375
990
1.848
12.300
14.240

• Find out two numbers with five digits that are divisible by both 2 and 5 and aren’t divisible by 100
• Write down two numbers with five digits that are multiples of:

a) 3 and 11 but not of 9

b) 9 and 11. Are they multiples of 3?

4.- PRIME FACTOR DECOMPOSITION OF A NUMBER

5.- THE HIGHEST COMMON FACTOR AND THE LEAST COMMON MULTIPLE

5.1.- Concept of the highest common factor (HCF)

Definition:

The highest common factor of several numbers is the largest number that evenly divides into all of them.

10.2.- Rule for calculating the h.c.f

Regla:

“To work out the hcf of several numbers, first you have to find the prime factor decomposition of the given numbers and then, to take the common factors with the least index”.

Solve the following exercises:

• Work out the factors of the numbers below and then find out the hcf:

a) 2 and 16                     b) 3 and 25                  c) 9, 12 and 18               d) 27, 36 and 63

• Find out the hcf of the following numbers using the Spanish and the English methods:

a) 4, 6, 18 and 32                    b) 3, 4, 12, 36 and 48

5.3.- Concept of the least common multiple (lcm)

Definition: The least common multiple of several numbers is the smallest number that is multiple of all of them.

5.4.- Rule for calculating the lcm

Regla:

“To work out the lcm of several numbers, first write them as a product of their prime factors and then take the common and non-common factors with the highest index.”

Solve the following exercises:

• Work out the l.c.m. of the numbers below:

a) 9, 12 and 18            b) 27, 36 and 63

• Work out the l.c.m. of the following numbers. What conclusion do you reach?

a) 2, 4, 8 and 16          b) 3, 4, 6 and 12.

• Do this test, don´t copy anything.  Call your teacher once you have finished your test:

Juega con las matemáticas:

1.-The decimal numeral system.

1.1.-The natural number set.

Natural numbers are the naumbers used for counting things. Natural numbers are positive numbers (numbers taht more than 0). They are 1, 2, 3, 4, … and so on until infinity. Natural numbers have two main purposes: you can use them for counting and for ordering. Mathematicians use N to refer to the set of all natural numbers.

Natural numbers are also called Counting Numbers. In Spanish they are called números cardinales if they are used for counting or números ordinales if they are used for ordering.

1.2.-The decimal numeral system.

Un sistema de numeración es el conjunto de reglas y símbolos que hacen posible la representación de números.

The Sumerians were the first people with a numeral system. Since then, Egyptians, Mayas, Romans, etc. have had their own numeral system. But around 773, a positional system began in India. Then Indian system transmitted to Europe by Arabs, so it was

Nuestro sistema de numeración es decimal y posicional. Usa diez símbolos diferentes (0, 1, 2, 3, 4, 5, 6, 7, 8 y 9) y además, el valor de cada cifra depende de su posición. Por ejemplo, en el número 21.320, el primer 2 tiene una valor de 20.000 unidades, sin embargo el último 2 tiene un valor de 20 unidades.

Definition:  Our numeral system is a decimal and a place-value notation system. It is decimal because it is formed with ten symbols and it is positional because the value of each digit depends on the position in the number.

En el sistema decimal cada posición representa una potencia de 10. Así, empezando por la derecha, la primera posición equivale a 1 unidad, la segunda a 10 unidades, la tercera a 100, etc. Cada posición recibe un nombre en función de su valor:

Exercise: Write a number with 8 digits over the lines and translate the following words:

___ ___ ___ ___ ___ ___ ___ ___

                                                                   Units or Ones:__________
                                                                   Tens : __________
                                                                   Hundreds: ___________
                                                                   Thousands:_____________
                                                                   Ten thousands: __________
                                                                   Hundred thousands: ______
                                                                   Millions:_______________
                                                                   Ten millions: ___________

Un número puede descomponerse en suma de productos que expresen el valor de cada una de sus cifras en función de su posición o al revés:

Solved example:

1.- Write the number 12.034.152 as an addition

12.034.152 = 1×10.000.000 + 2×1.000.000 + 3×10.000 + 4×1.000 + 1×100 + 5×10 + 2×1=

In Spanish: 1 Decena de Millón + 2 Unidades de Millón + 3 Decenas de Millar + 4 Unidades de Millar + 1 Centena + 5 Decenas + 2 Unidades

In English: 1 Ten Millions + 2 Millions + 3 Ten Thousands + 4 Thousands + 1 Hundreds + 5 Tens + 2 Units

2.- Find out the number represented by 3 Millions + 3 Hundred Thousands + 12 Thousands +35 Tens + 2 Units.

Hay dos formas de hacer este ejercicio:

Calculando el valor de cada cifra según su posición y sumando:

3 Millions = 3 x 1.000.000 =                3.000.000

3 Hundred Thousands = 3 x 100.000 = 300.000

12 Thousands = 12 x 1.000 =                     12.000

35 Tens = 35 x 10 =                                           350

2 Units = 2 x 1 =                                                      2

                                                                    3.312.352

Dibujando líneas con los nombres de las posiciones y colocando las cantidades una por una. Si una cantidad tiene más de una cifra entonces hay que escribirla empezando en su correspondiente posición pero hacia la izquierda.

___ ___ ___ . ___ ___ ___. ___ ___ ___
HM  TM  UM    HT   TT   UT      H      T     U

Solve the following exercises:

1A.- Write the numbers below as an addition in English and in Spanish like the solved example above.

            a) 12.004.23;                                                        b) 103.245.023;

2A.- Find out the numbers which are formed by the following values:

a) 13 Unidades de Millón + 52 Unidades de Millar + 3 Centenas + 24 Unidades =

b) 12 Centenas de Millón + 124 Decenas de Millar + 34 Decenas + 5 Unidades =

c) 2 Thousand Millions + 3 Millions + 5 Ten Thousands + 2 Thousands + 3 Hundreds + 5Units=

d) 3 Hundred Millions + 12 Millions + 134 Hundreds + 12 Units =

1.3.- Representation and ordering

The natural numbers can be represented on a half line (semirrecta) (line with a fixed beginning and with no fixed ending) that begins with zero and which is divided in equal segments.
0 1 2 3 4 5 6 7 8
Esta representación puede usarse para ordenar los números. Un número es mayor que otro si está situado más a la derecha en la semirrecta.
To compare two numbers, we can use three symbols: > (greater than: mayor que), = (equal to: igual a) ; < (less than: menor que).

Solve the exercise:

3A.- Put the corresponding symbols between the following numbers:
5 ___ 7 1003 ____1030 10020 ___10200 9898____ 9799

2.- READING AND WRITING NUMBERS

2.1.- Describing a number in words.

Para leer un número la forma más fácil es usar los separadores de miles cada tres cifras empezando por el final. Después nombraremos los puntos (mil, millón, mil, billón, etc.). Para leer el número iremos leyendo cada grupo de tres cifras y a continuación el nombre del punto.

Solved example:

3.- Describe in words the number 12.045.235.003.134:

· Primero nombraremos cada uno de los puntos del número:

12.045.235.003.134

Billion     Thousand        Million    Thousand

  Billón                  Mil         Millón                 Mil

· Ahora leeremos cada grupo de tres números y después el nombre del punto:
Twelve billion forty five thousand two hundred and thirty five million three thousand one hundred and thirty four.
Doce billones cuarenta y cinco mil doscientos treinta y cinco millones tres mil ciento treinta y cuatro.
(Note: In English, numbers are written using commas instead of dots.
Example: 12,045,235,003,134)

Solve the exercise:

4A.- Describe the following numbers in words, in Spanish and in English:

a) 34.000.340.02: b) 3.004.000.123.004:

c) 12.005.000.012.300: d) 373.005.000.000.345

2.2.- Writing an amount in figures:

Para escribir una determinada cantidad en cifras numéricas subraya todas las palabras que hagan referencia al nombre de un separador de miles (millón, mil, etc.). Escribe los puntos separados por un espacio, deberás de empezar por el mayor que aparezca y escribirlos todos hasta llegar al separador de mil. Por último escribe entre los puntos los grupos de números completando hasta tres cifras en cada caso.

Solved example:

4.-Write ten billion one hundred thirteen million two thousand twenty three in figures:

· Primero subrayaremos todas las palabras que hagan referencia a mil, millón, billón, etc:
ten billion one hundred thirteen million two thousand twenty three
diez billones ciento trece millones dos mil veintitres

· Ahora tenemos que escribir empezando por el mayor.(Hay que escribirlos todos aunque no se nombren en el número)

. . . .

Billion Thousand Million Thousand

Billón Mil Millón Mil

· Tenemos que escribir las cantidades entre los puntos, siempre completando con ceros para que haya tres cifras entre cada dos puntos:
                                                                10.000.113.002.023

Solve the exercise:

5A.- Write the following amounts in figures:
a) Cuarenta billones tres mil millones ciento cuarenta mil.
b) Trece billones, doscientos mil tres millones ciento doce mil cuatro.
c) Twelve billion two hundred and forty thousand million three hundred thousand and five.
d) Four billion twenty seven thousand two million five thousand three hundred and fifteen.
e) Fifteen billion forty five thousand three hundred and twenty four, six million and two hundred.

3.- OPERATIONS WITH NATURAL NUMBERS

3.1.- Addition

Adding is the same as putting together or joining two values into one. Es reunir, juntar, añadir. We read 3 + 5 = 8 like: “Three plus five is equal to eight” or “Three plus five equals eight” or “Three plus five is eight”. Terms in the addition are called addends and the result is called the sum. In Spanish the addends are the sumandos.

Solved Example:

5.- The library has lent 45 books last Monday, 50 books on Tuesday and 73 books on Wednesday.
How many books have they lent?
45 + 50 + 73 = 168 books. Answer: They have lent 168 books.

3.2.- The properties of addition.

The properties are the closure, commutative, associative, and additive identity:
Closure property: Addition of two natural numbers is always another natural number. For example 6 + 7 = 13
Commutative property: When two natural numbers are added, the sum is the same regardless of the order of the addends. a+ b = b + a. For example 4 + 2 = 2 + 4
Associative Property: When three or more natural numbers are added, the sum is the same regardless of the grouping of the addends. a + b + c = (a + b) + c = a + (b + c). For example (2 + 3) + 4 = 2 + (3 + 4)
Additive Identity Property: The sum of any natural number and zero is the original number. For example 5 + 0 = 5.

Exercise : (Try to translate the properties into Spanish)

Ley de Composición interna: ________________________________________________________________ Propiedad Conmutativa:
________________________________________________________________ Propiedad Asociativa:
________________________________________________________________ Elemento neutro:
________________________________________________________________

 3.3.- Subtraction of natural numbers.

Subtracting is removing or taking away some objects from a group. Es quitar, eliminar. We read 13 – 7 = 6 like: “Thirteen subtract seven equals six” (sometimes you can see “thirteen take away seven equals six” but it is better to use the first expression. The terms of subtraction are called minuend and subtrahend, the outcome is called the difference.

The minuend is the first number, it is the number from which you take something and it must be the larger number. In Spanish it is called minuendo.
The subtrahend is the number that is subtracted and it must be the smaller number. In Spanish it is called sustraendo.
The difference is the result of the subtraction. In Spanish it is called diferencia.
To check if the subtraction is correct we add up the subtrahend and the difference. The outcome must be the minuend.

Prueba de la resta:
Minuend = Subtrahend + Difference;
And in Spanish: Minuendo = Sustraendo + diferencia

Solved example:

6.- We have saved 3520 euros but we have spent € 745 on a computer. How much money is left?
                         3520 – 745 = 2775.
Answer: 2775 euros is left.

3.4.- Multiplication

Multiplying is doing an addition of equal addends. Es hacer una suma de sumandos iguales.
                 3 + 3 + 3 + 3 + 3 = 3 x 5 = 15
We read 3 x 5 = 15 like: “Three times five equals fifteen” or “Three times five is fifteen.” The factors are the numbers that are multiplied together. The product is the result of multiplying.

Solved Example:

7.- In my living-room I have a bookcase with three shelves.  If there are five books on each shelf, how many books are there?
                                                                   5 x 3 = 15
Answer: I have 15 books in my bookcase.

3.5.- The properties of multiplication

The properties are the closure, commutative, associative, and additive identity.

Closure property: Multiplication of two natural numbers is always another natural number.For example 6 x 7 = 42
Commutative property: When two numbers are multiplied together, the product is the same regardless of the order of the factors. For example 4 x 2 = 2 x 4
Associative Property: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example: (2 x 3) x 4 = 2 x (3 x 4)
Multiplicative Identity Property: The product of any number and one is that number. For example 5 x 1 = 5.

Definition : (Try to translate the properties into Spanish)

Ley de Composición interna: ________________________________________________________________ Propiedad Conmutativa:________________________________________________________________ Propiedad Asociativa:________________________________________________________________ Elemento neutro:________________________________________________________________

3.6. Division

Dividing is to share a quantity into equal groups. Es repartir en partes iguales. It is the inverse of multiplication. In Spanish we write 6 : 2 , but in English it is always 6 ÷ 2 and never with the colon (:).
We read 15 ÷ 5 = 3 like: “Fifteen divided by five equals three”.
There are four terms in a division: dividend, divisor, quotient and remainder:
The dividend is the number that is divided. In Spanish is dividendo.
The divisor is the number that divides the dividend. In Spanish is divisor.
The quotient is the number of times the divisor goes into the dividend. In Spanish is cociente.
The remainder is a number that is too small to be divided by the divisor and in Spanish is called resto.

Solved Example:

8.- There are 72 sweets in a bag. If we want to distribute them to 12 children, How many sweets are there for each child?
                                                        72 : 12 = 6
Answer: Six sweets for each child.

La división puede ser:

a) Exacta: Tiene resto cero.

b) Entera: Tiene resto distinto de cero.

To check if the division is correct we do the division algorithm (prueba de la división):

Division Algorithm:                                        Dividend = Divisor x Quotient + Remainder;

And in Spanish:                                              Dividendo= Divisor x Cociente + Resto.

Solved example:

9.- Find out the outcome of the division 237 : 13 and then check the result with the division algorithm:
                                  237 : 13 = 18      Remainder = 3

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

4. COMBINED OPERATIONS

4.1.- Distributive property:

La suma de dos números multiplicada por un tercero es igual a la suma del producto de cada término de la suma por el tercer número.

For example 4 x (6 + 3) = 4 x 6 + 4 x 3.

Así que para hacer la multiplicación de un número por un paréntesis que tiene una suma: 
First, the brackets and then the multiplication:   12 x ( 3 + 5 ) = 12 x 8 = 96
Applying the distributive property:                      12 x ( 3 + 5 ) = 36 + 60 = 96

Solved example:

10.- Do the operation 5 x (12 + 45) in two different ways:
5 x (12 + 45 ) = 5 x 57 = 285 First, the brackets
5 x (12 + 45) = 60 + 225 = 285 Applying the distributive property

Solve the exercise:

6A.- Do the following operations in two different ways:

a) 12 x (12 + 4) =

b) 3 x (2 + 1 + 7) =

c) (12 + 30) x 5

4.2.- Order of the operations

When expressions have more than one operation, we have to follow rules for the order of operations:

• Regla 1: Primero se hace cualquier operación entre paréntesis.

• Regla 2: Después multiplicaciones y divisiones, de izquierda a derecha.

• Regla 3: Por último sumas y restas, de izquierda a derecha.

  To remind this you can use the PEMDAS rule:

  P: Parenthesis.
  E: Exponents.
  M: Multiplications.
  D: Divisions .
  A S: Additions and subtractions.

Solved Examples:
11.- Solve 3 + 6 x (5 + 4) 3 - 7 using the order ÷ of operations.
Step 1: 3 + 6 x (5 + 4) ÷ 3 - 7 = 3 + 6 x 9 ÷ 3 - 7 Brackets
Step 2: 3 + 6 x 9 ÷ 3 - 7 = 3 + 54 ÷ 3 - 7 Multiplication
Step 3: 3 + 54 ÷ 3 - 7 = 3 + 18 - 7 Division
Step 4: 3 + 18 - 7 = 21 - 7 Addition
Step 5: 21 - 7 = 14 Subtraction

12.- Evaluate 9 - 5 ÷ (8 - 3) x 2 + 6 using the order of operations.
Step 1: 9 - 5 ÷ (8 - 3) x 2 + 6 = 9 - 5 ÷ 5 x 2 + 6 Brackets
Step 2: 9 - 5 ÷ 5 x 2 + 6 = 9 - 1 x 2 + 6 Division
Step 3: 9 - 1 x 2 + 6 = 9 - 2 + 6 Multiplication
Step 4: 9 - 2 + 6 = 7 + 6 Subtraction
Step 5: 7 + 6 = 13 Addition

Como ves en los ejemplos anteriores las multiplicaciones y divisiones o las sumas y las restas se van realizando de izquierda a derecha, nunca de dos en dos. Si dentro de un paréntesis hay varias operaciones volveremos a aplicar la regla PEMDAS a su vez dentro del paréntesis como se ve en el siguiente ejemplo:

13.- Evaluate 150 ÷ (6 + 3 x 8) - 5 using the order of operations.

Solution:

Step 1: 150 (6 + 3 x 8) - 5 = 150 (÷ ÷ 6 + 24) - 5 Multiplication inside brackets.
Step 2: 150 ÷ (6 + 24) - 5 = 150 ÷ 30 - 5 Addition inside brackets.
Step 3: 150 ÷ 30 - 5 = 5 - 5 Division.
Step 4: 5 - 5 = 0 Subtraction.

Solve the exercise:

7A.- Solve using the order of operations:

a) 5 + 2 x (10 – 2 x 5 + 1) – 3 =

b) 10 – 3 x 2 + 35 : (5 – 4 + 3 x 2) =

5. POWERS AND ROOTS.

5.1.-Index form

The notation 3² and 2³ is known as index form. The small digit is called the index number or power. You have already seen that 3² = 3 × 3 = 9, and that 2³ = 2 × 2 × 2 = 8. Similarly, 5⁴ (five to the power of 4) = 5 × 5 × 5 × 5 = 625
and 3⁵ (three to the power of 5) = 3 × 3 × 3 × 3 × 3 = 243. The index number tells you how many times to multiply the numbers together.

• ·   When the index number is two, the number has been ‘squared‘.
• ·   When the index number is three, the number has been ‘cubed‘.
• ·   When the index number is greater than three you say that it is has been multiplied ‘to the power of‘.

For example: 7² is ’seven squared’,
3³ is ‘three cubed’,
3⁷ is ‘three to the power of seven’,
4⁵ is ‘four to the power of five’.

Question
Look at the table and work out the answers. The first has been done for you.

43	4 × 4 × 4	64
27	2 × 2 × 2 × 2 × 2 × 2 × 2
72	7 × 7
53
24
65

5.2 Multiplication

How can we work out 2³ × 2⁵
2³ = 2 × 2 × 2
2⁵ = 2 × 2 × 2 × 2 × 2
so 2³ × 2⁵ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸

There are 3 twos from 2³ and 5 twos from 2⁵, so altogether there are 8 twos. In general, 2^m × 2ⁿ =2^{(m + n)}

Solved Examples:2⁵ × 2⁴ = 2^{(5 + 4)} = 2⁹ 2⁷ × 2³ = 2^{(7 + 3)} = 2¹⁰
The rule also works for other numbers, so 3⁴ × 3² = 3^{(4 + 2)} = 3⁶
25⁶ × 25⁴ = 25^{(6 + 4)} = 25¹⁰

5.3.-Division

If you divide 2⁵ by 2³ you see that some of the 2’s cancel:

After division two 2s are left. 
Five 2s are divided by three 2sSo 2⁵ ÷ 2³ = 2²
In general, 2^m ÷ 2ⁿ = 2^{(m - n)}

Solved Example.-
2⁵ ÷ 2² = 2^{(5 - 2)} = 2³ 2⁷ ÷ 2³ = 2^{(7 - 3)} = 2⁴
The rule also works for other numbers, so 5¹⁰ ÷ 5³ =5^{(10 - 3)} = 5⁷
45⁹ ÷ 45⁴ = 45^{(9 - 4)} = 45⁵

5.4 Roots

Square root

The opposite of squaring a number is called finding the square root.

Solved Examples:

The square root of 16 is 4 (because 4² = 4 × 4 = 16)

The square root of 25 is 5 (because 5² = 5 × 5 = 25)

The square root of 100 is 10 (because 10² = 10 × 10 = 100)

Solve:

What is the square root of 4?

The symbol ‘√ ‘ means square root, so
√ 36 means ‘the square root of 36′, and
√ 81 means ‘the square root of 81′

You will also find a square root key on your calculator.

Cube root

The opposite of cubing a number is called finding the cube root.

Solved Example:

The cube root of 27 is 3 (because 3 × 3 × 3 = 27)

The cube root of 1000 is 10 (because 10 × 10 × 10 = 1000)

Solve:

What is the cube root of 8?

Practica con la BBC . Tras la revision de contenidos, realiza el test que te proponen copiandolo en tu cuaderno.

1.- THE DECIMAL NUMERAL SYSTEM. HOW TO WRITE NUMBERS

Nuestro sistema de numeración es DECIMAL Y POSICIONAL:

DECIMAL porque se compone de diez símbolos: 0, 1, 2, 3, ……… y diez unidades del mismo orden forman una unidad del orden superior.

1 decena = 10 unidades

1 centena = 10 decenas = 100 unidades

1 unidad de millar = 10 centenas = ……. = 1000 unidades

1 decena de millar = 10 unidades de millar = ……… = 10.000 unidades

1 centena de millar = 10 decenas de millar = ……… = 100.000 unidades

1 unidad de millón = 10 centenas de millar = ……. = 1.000.000 unidades

POSICIONAL porque el valor de cada símbolo depende de la posición en el número.

In English, the names of the place values are:

1 ten = 10 units

1 hundred = 10 tens = 100 units

1 thousand = 10 hundreds = ……. = 1000 units

1 ten-thousands = 10 thousands = ……… = 10.000 units

1 hundred-thousands = 10 ten-thousands = ……… = 100.000 units

1 million = 10 hundred-thousands = ……. = 1.000.000 units

Exercise 1. You must do only the first block of exercises. Read the instructions carefully.

When you have finished, copy the last example into your notebook.

Exercise 1. Example: …..

Exercise 2. Copy two orders and your solutions into your notebook.

Exercise 2. Order:…. Solution: …..

This blog is for the students of the bilingual section at “Virgen de Covadonga” Secondary School, in El Entrego, Asturias, Spain. They are studying their first year of their secondary compulsory education (1º ESO). They are 11-12 years old. This schoolyear they are having part of their Maths  in English. This blog is for the activities of maths.