Global Thinkers: Mathematics 2. Secondary (sample)

What are we going to learn?

BLOCK I

II

BLOCK III

BLOCK IV

•

•

•

V

• Trial and error

• Work systematically

• Lowest common multiple

• Greatest common divisor

• The Z set of integers

• Square root of a decimal number

• Fractions

• Solving problems with fractions

• Powers and fractions

• Inversely proportional magnitudes

• Solving problems of compound proportionality

• Bank interest

• Other arithmetic problems

• Operating with integers

• Powers of integers

• Square roots of integers

• Fractions and decimal numbers

• Solving problems of proportional distribution

Analyse how best to fill a space with objects of known dimensions.

Choose a food product and prepare an economic analysis.

Analyse experts' recommendations about how much sleep we need.

• Polynomials

• Notable products

• Solving simple equations

• Equations with denominators

• General method for solving first-degree equations

• Methods for solving linear systems

• Solving problems with equations

• Second-degree equations

• Solving second-degree equations

• Solving problems using systems of equations

Calculating the approximate cost of sending a packet.

Use percentages in real-life situations.

Solve a problem using algebraic expressions.

Use algebraic language.

Invent a problem and solve it with a system of equations.

• Applications of the Pythagorean theorem

• How to build similar shapes

• Thales' theorem

• Regular polyhedra

• Cross-sections of polyhedra

• Cylinders

• Volume of a prism and a cylinder

• Volume of a pyramid and a truncated pyramid

• Functions given as equations

• Proportional functions: y = mx

• The slope of a line

• Assigning probabilities to regular experiments

• Similarity between right-angled triangles

• Applications of the similarity of triangles

• Cones

• Truncated cones

• Spheres

• Volume of a cone and a truncated cone

• Volume of a sphere

• Linear functions: y = mx + n

• Constant functions: k

• Some strategies for calculating probability

How far is the horizon?

Measure distances and hights in the countryside.

Let's make solids of revolution.

Let's estimate the costs of restoring the chapel.

Journeys and diving trips: describe them using graphs.

Let's calculate conditional probability!

You are going to analyse a packing and storage problem. You will explore the different options and describe the pros and cons of each. Then you will justify the optimal solution you choose.

1. How many ways can he pack the cheeses?

2. How many cheeses fit on each shelf, depending on how he puts them?

3. Which is the best way to use the space?

4. Which is the best option, in your opinion? Include the pros and cons.

5. Can you think of a better size (or shape) for the shelves or the cheeses?

6. If the cheeses were shaped like cubes, what would be the ideal size?

What are yougoing to learn to

TAKE ACTION?

THE SET OF NATURAL NUMBERS

• Decimal numeral system and the value of the digits of a number

THE Z SET OF INTEGERS

• Adding and subtracting integers

OPERATING WITH INTEGERS

• Adding and subtracting inside brackets

• Combined operations

Key language

Multiple of Divisor of

|a| absolute value of Raised to

Learning by doing videos 1-minute explanations videos

THE RELATION OF DIVISIBILITY

• Relation of divisibility

• Multiples of a number

• Divisors of a number

GREATEST COMMON DIVISOR

• Greatest common divisor

POWERS OF INTEGERS

• Multiplication with integers

• Powers of negative numbers

• Decomposing a number into prime factors

PRIME AND COMPOSITE NUMBERS

• Eratosthenes' sieve

LOWEST COMMON MULTIPLE

• Lowest common multiple

SQUARE ROOTS OF INTEGERS

• Other roots

• Perfect square and cube

• The properties of powers

MY MATHS DICTIONARY

In context

|–8| = 8 The absolute value of minus eight is eight.

5 raised to the power of 4 is 625.

The set of natural numbers

The numbers we use to count are called natural numbers

• The letter N represents the set of natural numbers (it has a beginning and order, but no end):

N = {0, 1, 2, 3, 4, …}

Natural numbers

• We can represent and order them on a number line:

The decimal numeral system

In human history, different cultures have created and used different numeral systems (ways of expressing natural numbers).

Some numeral systems

We use the decimal numeral system (DNS). Ancient Indian mathematicians invented it. The Arabs brought it to the Mediterranean in the 8th century.

The decimal numeral system is a positional system:

Mayan numerals

Each digit has a place value.

It is a decimal system because ten units in every place value is equivalent to one unit of the place value immediately to the left.

1 HTh = 10 TTh = 100 Th = 1 000 H = 10 000 T = 100 000 U

We can decompose a number like this (polynomial decomposition):

We use base 10 in the decimal numeral system because of the primitive method of using our fingers to count.

The sexagesimal system

Some countries in human history counted in 60s. This is called the sexagesimal system. Why did they adopt this system?

One hand for keeping count.

5 x 12 = 60

We use the sexagesimal system to measure time and angles.

One hand for counting the 12 finger bones of the index, middle, ring and little finger (using the thumb as a guide).

Each place value is divided into 60 units of the place value immediately to the left.

The symbols we use for the minutes and seconds are different for each magnitude.

When we measure an amount, we can express the magnitude in two ways

Theory into practice

Copy and complete in your notebook.

1 Express in complex form. a)

2 Express 2 hours and 24 minutes in simple form (first in minutes and then in seconds).

a) In minutes: 2 h 24 min 8 (2 · 60 + 24) min = (… + …) min = … min

b) In seconds: 2 h 24 min 8 (2 · 3 600 + 24 · …) s = … s

The relation of divisibility 2

Multiples and divisors

Two numbers have a relationship of divisibility when their quotient is exact.

If the division a : b is exact

a is a multiple of b.

b is a divisor of a.

The multiples and divisors of a number

The multiples of a number contain that number an exact number of times. They are the result of multiplying that number by another natural number.

Some multiples of 12

Every number is a multiple of itself and one.

The divisors of a number are contained in that number an exact number of times. They divide it with an exact quotient. A number has a finite number of divisors.

A number has an infinite number of multiples.

A number has at least two divisors: itself and one.

Keep in mind

•

a multiple of every number.

• Zero has only one multiple: itself.

A property of multiples

If we add up two multiples of 12, we get another multiple of 12:

36 + 60 = 12 · 3 + 12 · 5 = 12 · (3 + 5) = 12 · 8 = 96

The sum of two multiples of a number, a, is another multiple of a.

m · a + n · a = (m + n) · a

Divisibility criteria

These are a set of practical rules to quickly check if a number is a multiple of 2, 3, 5, 11…

Divisibility rules for 2, 5 and 10

• A number is a multiple of 2 if it ends in 0, 2, 4, 6 or 8.

• A number is a multiple of 5 if it ends in 0 or 5.

• A number is a multiple of 10 if it ends in 0.

Divisibility rules for 3 and 9

• A number is a multiple of 3 if the sum of its digits is a multiple of 3. We can decompose a number with several digits into a multiple of 3 plus the sum of its digit:

We can decompose a number with several digits into a multiple of 2 plus the units digit:

The first addend is a multiple of 3. A number is a multiple of 3 when the second addend is also a multiple of 3.

• A number is a multiple of 9 if the sum of its digits is a multiple of 9. The reasoning above also applies for multiples of 9.

• A number formed by nines is a multiple of 3 and 9.

Divisibility rules for 11

• A number is a multiple of 11 if the sum of the digits in the odd position minus the sum of the digits in the even position (or vice versa), is a multiple of 11.

• We can decompose a number with several digits into a multiple of 11 plus the result of adding and subtracting its digits alternatively:

649 = 600 = 594 + 6 40 = 44 – 4

The first addend is a multiple of 11. A number is a multiple of 11 when the second addend is also a multiple of 11.

Keep in mind

All the numbers in the boxes are multiples of 11. Check it!

Theory into practice

Copy and complete in your notebook. 1 Divide, look and answer.

a) Is 173 a multiple of 19? And 228?

b) Is 43 a divisor of 516? And of 743?

2 Write the first eight multiples of 13.

13

3 Look and write all the divisors of 42.

The multiples of 14 between 250 and 300 are:

Let’s practise!

5 Write.

a) The first five multiples of 20.

b) All the divisors of 20.

6 Draw all the ways of representing 36 as a rectangular number.

36 = 3 · 12

What is the relationship with the divisors of 36?

7 Write all the pairs of numbers with a product of 60.

8 Find.

a) All the multiples of 7 between 100 and 150.

b) The first multiple of 13 after 1 000.

9 Copy, circle the even numbers and cross out the multiples of 3.

45 - 67 - 74 - 96 - 143 - 138 - 251 - 309 – 488

10 What is the value of a in each case?

5 6 a

a) The number is a multiple of 2.

b) The number is a multiple of 3.

c) The number is a multiple of 5.

d) The number is a multiple of 9.

11 Which of the numbers below are multiples of 11?

286 611 913 1 804 2 444 3 333

12 Look, copy and complete in your notebook.

a) n = 2 · 3 · k = 6 · k → If a number, n, is a multiple of 2 and 3, it is also a multiple of 6.

b) m = 2 · 5 · k = 10 · k → If a number, m, is a multiple of 2 and 5, it is also a multiple of ...

c) p = 15 · k = 3 · 5 · k → If a number, p, is a multiple of 15, it is also a multiple of ... and ... .

Prime and composite numbers

There are two types of numbers depending on the number of factors they have:

We cannot decompose a prime number into factors. They only have two divisors: the number and one. For example:

13 = 13 · 1

The prime numbers smaller than

We can decompose a composite number into a product of factors.

For example:

40 = 8 · 5 = 2 · 2 · 2 · 5

Decomposing a number into its prime factors

You reach the highest level of a number’s factorial decomposition when all the factors are prime numbers.

To decompose a number into prime factors, we do it systematically. Look at the decomposition into prime factors of 594:

To decompose a number into prime factors, think about the divisibility criteria.

1 Classify as prime numbers or composite numbers.

3 Decompose these numbers into prime factors.

a) 84 b) 130 c) 160 d) 594

e) 720 f ) 975 g) 2 340 h) 5 220

4 Factorise without doing any operations.

a) Three multiples of 12 = 22 · 3.

b) All the divisors of 75 = 3 · 5 · 5.

5 m = 22 · 3 · 5 and n = 23 · 3. Write the numbers.

a) Three common multiples of m and n

b) Three common divisors of m and n.

Lowest common multiple 4

The lowest common multiple (LCM) of a set of numbers, a, b, c, … is the smallest of their common multiples. We write it like this:

LCM (a, b, c, …)

Calculating the lowest common multiple

To calculate the lowest common multiple of a set of numbers:

• First, we decompose the numbers into prime factors. For example, to calculate the LCM of 200 and 240:

200 = 23 · 52 240 = 24 · 3 · 5

• Then, we take all the prime factors (common and not common) raised to the highest exponent that appears.

The lowest common multiple of 200 and 240 is 1 200:

LCM (200, 240) = 1 200

A factory manufactures sports shoes and boots. The sports shoes have 100 cm laces (200 cm per pair) and the boots 120 cm laces (240 cm per pair).

The factory wants to buy a long piece of shoelace to make an exact number of sports shoes and boots. What length of shoelace must they buy?

LCM (200, 240) = 1 200

Solution: They must buy 1 200 cm (12 m) or a multiple of that amount: 12 - 24 - 36… metres.

Let’s practise!

1 Calculate. Use mental arithmetic.

a) LCM (3, 5)

b) LCM (6, 11)

c) LCM (10, 15) d) LCM (10, 25)

e) LCM (30, 40) f) LCM (50, 100)

2 Calculate.

a) LCM (18, 24)

b)

LCM (21, 35)

c) LCM (72, 90) d)

e) LCM (60, 72, 90) f)

LCM (90, 120)

LCM (50, 75, 100)

3 A supermarket does an inventory every 36 days and reorganises the shelves every 24 days. How often do both events happen on the same day?

4 A gear system has two gears, one with 24 teeth and the other, 32. How many times does each gear rotate before they are both back in their original position?

Greatest common divisor

The greatest common divisor (GCD) of a set of numbers, a, b, c, … is the largest of their common divisors. We write it like this:

GCD (a, b, c, …)

Calculating the greatest common divisor

To calculate the greatest common divisor of a set of numbers:

• First, we decompose the numbers into prime factors. For example, to calculate the GCD of 200 and 240:

200 = 23 · 52 240 = 24 · 3 · 5

• Then, we take the common prime factors, each raised to the smallest exponent that appears.

200 = 2 · 2 · 2 · 5 · 5

240 = 2 · 2 · 2 · 2 · 3 · 5

8 GCD (200, 240) = 23 · 5 = 40

The greatest common divisor of 200 and 240 is 40:

240) = 40

Problem solved

A carpeter wants to cut two pieces of wood measuring 200 cm and 240 cm into equal pieces that are as long as possible, without any remaining wood.

How long does he cut each piece?

GCD (200, 240) = 40

Solution: He cuts each piece 40 cm long.

Let’s practise!

1 Calculate. Use mental arithmetic.

a) GCD (4, 6)

b) GCD (6, 8)

c) GCD (5, 10)

d) GCD (15, 20)

e) GCD (18, 24)

f) GCD (50, 75)

2 A gardener has a piece of land measuring 248 cm × 250 cm. She wants to divide it into equal squares that are as big as possible, to plant flowers. What are the dimensions of the squares?

3 Calculate.

a) GCD (24, 36) b) GCD (28, 42)

c) GCD (63, 99) d) GCD (90, 126)

e) GCD (165, 275) f ) GCD (360, 450)

4 Take Action. We have a shelf that is 1.30 m long, 40 cm high and 20 cm deep. We want to put cubeshaped boxes on the shelf.

a) What are the maximum dimensions of these boxes?

b) What about boxes in other shapes and sizes? Find more than one answer.

The Z set of integers 6

The Z set of integers is formed by the N set of natural numbers, +a, and all the numbers in that set with a negative sign, –a, except for zero.

Absolute and opposite value of an integer

• Absolute value: the natural number without the sign before it. We write it like this: |a| → absolute value of a |+7| = 7 |–7| = 7

The absolute value of a number is equal to its distance from zero on the number line:

• Opposite of an integer: another integer with the same absolute value but with the opposite sign.

Opposite of (+7) → (–7)

Order of the Z set

Opposite of (–7) → (+7)

We can represent integers in order on a number line: …

Look at the number line. Note that:

• A number is bigger than all the numbers on the left and smaller than all the numbers on the right.

• All positive numbers are greater than zero and zero is greater than all negative numbers: (–7) < 0 < (+1).

• Negative numbers go ‘backwards’.

• In a group of negative numbers, the biggest number is the one with the smallest absolute value: (–12) < (–9) < (–2).

Let’s practise!

1 Write the absolute and the opposite value of each number.

a) –3 b) +8 c) –11

d) +23 e) –37 f ) +60

2 Order these numbers from smallest to biggest.

–7, –13, +8, –1, +1, +5, 0, +10, –24

3 True or false? Discuss.

a) All integers are also natural numbers.

b) All natural numbers are integers.

c) Only negative numbers have opposites.

d) Two opposite integers have the same absolute value.

Operating with integers 7

Adding and subtracting integers

This is how we add or subtract two integers:

• With the same sign: add their absolute values and use the same sign.

+4 + 7 = +11 –6 – 3 = –9

• With different signs: subtract their absolute values and use the sign of the number with the biggest absolute value.

–4 + 10 = + 6 +3 – 8 = –5

When we want to operate with more than two integers, we have two options:

• Operate step by step in the order that the integers appear.

8 – 10 + 6 – 5 – 3 = –2 + 6 – 5 – 3 = +4 – 5 – 3 = –1 – 3 = – 4

• Group the positive numbers on one side and the negative numbers on the other side. Then, calculate.

8 – 10 + 6 – 5 – 3 = 8 + 6 – 10 – 5 – 3 = +14 – 18 = – 4

Theory into practice

1 Read, think and complete in your notebook.

a) I receive €5 and then another €3. Now I have €8 more than before: +5 + 3 = …

c) I receive €10 and then I spend €3. Now I have €… … than before: +10 – 3 = …

b) I spend €4 and then another €2. Now I have € less than before: –4 – 2 = …

d) I receive €3 and then I spend €7. Now I have €… … than before: +3 – 7 = …

2 Copy and complete. Solve the same expressions in two different ways.

3 – 7 – 5 + 8 = – 5 + 8 = + 8 = … 3 – 7 – 5 + 8 = 3 + 8 – 7 – 5 = +11 – = …

Let’s practise!

3 Calculate. Use mental arithmetic.

a) 5 – 7 b) 2 – 9 c) –1 – 9

d) –12 + 17 e) –22 + 10 f) –12 – 13

4 Solve.

a) 10 – 3 + 5 b) 2 – 9 + 1 c) 16 – 4 – 6

d) 7 – 10 – 3 e) –7 – 8 + 5 f) –5 + 8 + 4

g) –8 + 2 + 3 h) –1 – 2 – 3 i) –7 – 3 – 4

5 Calculate.

a) 3 – 7 + 2 – 5

b) 2 – 6 + 9 – 3 + 4

c) 7 – 10 – 5 + 4 + 6 – 1

d) – 6 + 4 – 3 – 2 – 8 + 5

e) 12 + 5 – 17 – 11 + 20

f) 16

Addition, subtraction and brackets

Let’s look at these statements about a bank account:

I deposit €25. Now I have €25 more in my account. +(+25) = +25

I withdraw €55. Now I have €55 less in my account. –(+55) = –55

Remember this when operating with integers and brackets:

I pay €18. Now I have €18 less in my account. +(–18) = –18

I cancel a €60 bill. Now I have €60 more in my account. –(–60) = +60

• When there is a + sign before brackets, the signs inside the brackets do not change.

• When there is a – sign before brackets, the signs inside the brackets change: plus to minus, and minus to plus.

Here are two examples of operations with integers and brackets:

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

Theory into practice

6 Copy and complete. Solve the same expression in two different ways.

a) Removing the brackets first.

(7 – 10) – (2 – 5 + 4 – 9) = 7 – – 2 + – + = 7 + 5 + 9 – – – = 21 – = …

b) Doing the operations in the brackets first.

(7 – 10) – (2 – 5 + 4 – 9) = (–3) – ( – ) = (–3) – (– ) = …

Let’s practise!

7 Remove the brackets and calculate.

a) (–3) – (+4) – (–8)

b) –(–5) + (–6) – (–3)

c) (+8) – (+6) + (–7) – (–4)

d) –(–3) – (+2) + (–9) + (+7)

8 Solve by removing the brackets first.

a) (4 – 9) – (5 – 8)

c) 4 – (8 + 2) – (3 – 13)

e) 22 – (7 – 11 – 3) – 13

b) –(1 – 6) + (4 – 7)

d) 12 + (8 – 15) – (5 + 8)

9 Solve by doing the operations in the brackets first.

a) (2 – 6) + (4 – 8)

b) (8 – 10) – (12 – 7)

c) 15 – (2 – 5 + 8) + (6 – 9)

d) (8 – 6) – (3 – 7 – 2) + (1 – 8 + 2)

e) (5 – 16) – (7 – 3 – 6) – (9 – 13 – 5)

10 Solve in two ways, like in the example:

• 10 – (13 – 7) = 10 – (+6) = 10 – 6 = 4

• 10 – (13 – 7) = 10 – 13 + 7 = 17 – 13 = 4

a) 15 – (12 – 8) b) 9 – (20 – 6)

c) 8 – (15 – 12) d) 6 – (13 – 2)

e) 15 – (6 – 9 + 5) f) 21 – (3 – 10 + 11 + 6)

11 Calculate.

a) 7 – [1 + (9 – 13)] b) –9 + [8 – (13 – 4)]

c) 12 – [6 – (15 – 8)] d) –17 + [9 – (3 – 10)]

e) 2 + [6 – (4 – 2 + 9)] f) 15 – [9 – (5 – 11 + 7)]

12 Solve.

a) (2 – 9) – [5 + (8 – 12) – 7]

b) 13 – [15 – (6 – 8) + (5 – 9)]

c) 8 – [(6 – 11) + (2 – 5) – (7 – 10)]

d) (13 – 21) – [12 + (6 – 9 + 2) – 15]

e) [4 + (6 – 9 – 13)] – [5 – (8 + 2 – 18)]

f) [10 – (21 – 14)] – [5 + (17 – 11 + 6)]

Multiplying integers

Remember that a multiplication is a sum of identical addends:

(+3) · (– 6) = We add up (–6) three times:

+(– 6) + (– 6) + (– 6) = – 6 – 6 – 6 = –18

(–3) · (– 6) = We subtract (–6) three times:

–(– 6) – (– 6) – (– 6) = +6 + 6 + 6 = +18

RULE OF SIGNS

The product of two integers is:

• Positive if the factors have the same sign. (+) · (+) = + (–) · (–) = +

• Negative if the factors have different signs. (+) · (–) = –(–) · (+) = –

In these examples you can see how to apply the rule of signs:

• (+4) · (+3) = +12

• (+6) · (– 4) = –24

Dividing integers

• (–5) · (– 4) = +20

• (– 4) · (+8) = –32

The relationship between multiplication and division of integers is the same as with natural numbers.

(+4) · (+6) = +24 (+24) : (+4) = +6

(– 4) · (– 6) = +24 (+24) : (– 4) = – 6

(–24) : (+4) = – 6

(+4) · (– 6) = –24

As you can see on the right, the rule of signs also applies to the division of integers. (+)

(–24) : (– 6) = +4

13 Multiply.

a) (+10) · (–2) b) (– 4) · (–9)

c) (–7) · (+5) d) (+11) · (+7)

14 Look at the examples and multiply in two ways, as shown in the examples.

• (–3) · (+2) · (–5) = (– 6) · (–5) = +30

• (–3) · (+2) · (–5) = (–3) · (–10) = +30

a) (–2) · (–3) · (+4) b) (–1) · (+2) · (–5)

c) (+4) · (–3) · (+2) d) (– 6) · (–2) · (–5)

15 Divide.

a) (–18) : (+3) b) (–15) : (–5)

c) (+36) : (–9) d) (–30) : (–10)

e) (–52) : (+13) f ) (+22) : (+11)

16 Copy, complete and compare. What do you notice?

(+60) : [(–30) : (–2)] = (+60) : [+15] = [(+60) : (–30)] : (–2) = [ ] : (–2) =

17 Calculate the value of x in each case.

a) (–18) : x = +6 b) (+4) · x = –36

c) x · (–13) = +91 d) x : (–11) = +5

Combined operations

This is the order we follow to calculate the value of combined operations: Do

the brackets.

Theory into practice

18 Copy and complete. Find the value of the following expression:

(6 – 9 + 2) . (–5) + 3 . (2 – 6) + 4 = = ( ) (–5) + 3 ( ) + 4 =

= ( ) + ( ) + 4 = – + 4 = (6 – 9 + 2) · (–5) + 3 · (2 – 6) + 4

( ) (–5) + 3 · ( ) + 4

( ) + ( ) + 4 – + 4 =

Let’s practise!

19 Calculate like in the examples.

• 15 – 8 · 3 = 15 – 24 = –9

• 18 : 6 – 5 = 3 – 5 = –2

a) 18 – 5 · 3 b) 6 – 4 · 2

c) 7 · 2 – 16 d) 18 – 15 : 3

e) 5 – 30 : 6 f ) 20 : 2 – 11

20 Calculate like in the example.

• 21 – 4 · 6 + 12 : 3 = 21 – 24 + 4 = 25 – 24 = 1

a) 20 – 4 · 7 + 11 b) 12 – 6 · 5 + 4 · 2

c) 15 – 20 : 5 – 3 d) 6 – 10 : 2 – 14 : 7

e) 5 · 3 – 4 · 4 + 2 · 6 f ) 7 · 3 – 5 · 4 + 18 : 6

21 Look at the example and calculate.

• (–3) · (– 4) + (– 6) · 3 = (+12) + (–18) = 12 – 18 = – 6

a) 5 · (–8) – (+9) · 4

b) 32 : (–8) – (–20) : 5

c) (–2) · (–9) + (–5) · (+4)

d) (+25) : (–5) + (–16) : (+4)

e) (+6) · (–7) + (–50) : (–2)

f) (+56) : (–8) – (–12) · (+3)

22 Copy, calculate and complete in your notebook.

a) 18 – 5 · (3 – 8) = 18 – 5 · ( ) = 18 + =

b) 4 · (8 – 11) – 6 · (7 – 9) = 4 · ( ) – 6 · ( ) =

c) (4 – 5) · (–3) – (8 – 2) : (–3) =

23 Exercise solved

(– 2 ) · [ 11 + 3 · ( 5 – 7) ] – 3 · ( 8 – 11 ) –3 –2 – 6 +9 +5 –10 –1 (–2)

= (–2) · [11 – 6] + 9 = (–2) · [+5] + 9 = –10 + 9 = –1

24 Calculate.

a) 28 : (–7) – (–6) · [23 – 5 · (9 – 4)]

b) (–2) · (7 – 11) – [12 – (6 – 8)] : (–7)

Powers of integers 8

A power is a multiplication of identical factors.

Examples

• (+4)2 = (+4) · (+4) = +16

• (–3)4 = (–3) · (–3) · (–3) · (–3) = +81

• (–3)5 = (–3) · (–3) · (–3) · (–3) · (–3) = –243

You should keep in mind that: 10

Powers of negative numbers

When raising a negative number to a power

Even exponent = positive result

For example: (–3)2 = +9 (–3)4 = +81

Properties of powers

Odd exponent = negative result

For example: (–3)1 = –3 (–3)3 = –27

The following properties are fundamental to calculating powers.

Power of a product

The power of a product is equal to the product of the powers of the factors:

(a · b)n = an · bn

[(–2) · (+5)]3 = (–2)3 · (+5)3

[–10]3 (–8) · (+125)

–1 000 –1 000

Keep in mind that the power of an addition or subtraction is not equal to the sum of the powers of the addends:

[(–2) + (–3)]2 = [–5]2 = +25 (–2)2 + (–3)2 = 4 + 9 = +13

Power of a quotient

The power of a quotient is equal to the quotient of the powers of the dividend and the divisor:

(a : b)n = an : bn

[(–10) : (+5)]3 = (–10)3 : (+5)3

(–2)3 (–1 000) : (+125)

–8

–8

Product of powers with the same base

To multiply two powers with the same base, we add up the exponents:

am · an = am + n

(–10)2 · (–10)3 = (–10)2 + 3 = (–10)5

(+100) (–1 000)

–100 000 –100 000

Quotient of powers with the same base

To divide two powers with the same base, we subtract the exponents:

am : an = am – n

Power of another power

To raise one power to another power, we multiply the exponents:

(am)n = am · n

Let’s practise!

(–10)5 : (–10)3 = (–10)5 – 3 = (–10)2

(–100 000) : (–1 000)

+100 +100

[(–10)3]2 = (–10)3 · 2 = (–10)6

[–1 000]2

+1 000 000 +1 000 000

1 Write as a product when possible and calculate.

a) (–1)7 b) (–5)2 c) (–10)5

d) (–7)3 e) (–1)0 f) (–7)0

2 Calculate using a calculator, like in the example:

• 125 8 12**==== 8 {∫“¢°°«“} a) (–11)3 b) 175 c) (–27)4

3 Reduce to a single power like in the example:

• 25 · (–3)5 = [2 · (–3)]5 = (– 6)5

• (–15)4 : (+3)4 = [(–15) : (+3)]4 = (–5)4 = 54

a) 32 · 42 b) (–2)3 · 43

c) (+15)3 : (–5)3 d) (–20)2 : (– 4)2

4 Apply the property a m · a n = a m + n and reduce.

a) x 2 · x 3 b) a 4 · a 4 c) z 5 · z

5 Express as a single power.

a) (–2)5 · 27 b) (–2)3 · (+2)6

c) (–12)2 · (+12)2 d) (+9)4 · (–9)2

Keep in mind

6 Apply the property a m : a n = a m – n and reduce.

a) x 7 : x 4 b) a 7 : a 2 c) z 8 : z 3

7 Reduce to a single power.

a) (–7)8 : (–7)5 b) 109 : (–10)4

c) 124 : (–12) d) (– 4)10 : (+4)6

8 Apply the property (a m)n = a m · n and reduce.

a) (x 3)2 b) (a 3)3 c) (z 6)3

9 Copy and complete in your notebook.

a) (32)4 = 3 b) [(–2)4]3 = (–2)

c) [(+5)2]2 = (+5) d) [(– 6)3]5 = (– 6)

10 Reduce like in the example:

• (a 6 · a 4) : a 7 = a 10 : a 7 = a 3

a) (x 5 · x 2) : x 4 b) m 7 : (m 2 · m 3)

c) (a · a 6) : (a 2 · a 4) d) (z 5 · z 3) : (z 4 · z 2)

11 Calculate.

a) 106 : (54 · 24) b) (–12)7 : [(–3)5 · 45]

c) [(–9)5 . (–2)5] : 184 d) [57 · (– 4)7] : 204

Square roots of integers 9

The square root is the opposite of raising to the power of two (squaring):

a = b ï b 2 = a

The numbers that have an integer square root are called perfect squares.

The square roots of 16 are 4 and –4.

49 7 = ⇔ 749 2 = 400 20 = ⇔ 20 400 2 =

+4 8 because 42 = +16

– 4 8 because (– 4)2 = +16

Positive numbers have two square roots, one negative and one positive.

When we write 16 we refer to the positive root. When we write – 16 we refer to the negative root.

Take note:

() –16 = x ⇔ x 2 = –16 → It is impossible because the square of a number is never negative.

Negative numbes do not have a square root.

Other roots

A root index can be greater than two.

In general:

•

Let’s practise!

1 Calculate the square roots when possible.

a) () +1 b) () –1

c) () 25 +

d) () –36 e) () +100 f ) () –100

g) () –169 h) () 400 + i ) () –900

Take note

Every positive number has two fourth roots. The two fourth roots of 81 are:

() 81 4 + = +3 ï (+3)4 = +81

– () 81 4 + = –3 ï (–3)4 = +81

2 Think. Calculate when possible.

a) 27 3

b) –27 3

c) 16 4

d) –16 4 e) 32 5 f ) –32 4

g) –1 7 h) –1 8 i ) 64 6 +

My visual summary

Numeral systems

It is a positional system: each unit has a place value.

We use it to measure time and angles

Each unit is divided into 60 units of the place value immediately before. 1

}

We can use the decimal number system to write a number as large as we want.

If the division a : b is exact

a is a multiple of b.

b is a divisor of a.

If the sum of the digits in the odd position minus the sum of the digits in the even position is a multiple of 11.

A number is a multiple of:

You can factorise them. You cannot factorise them. They only have two divisors: themselves and one.

Factorising a number means decomposing it into prime factors.

Lowest common multiple

The lowest common multiple of several numbers is the smallest of their common multiples. We usually write it like this: LCM (a, b, c, ...).

1st We factorise. 2nd We take the common and not common factors raised to the highest exponent.

Greatest common divisor

The greatest common divisor of several numbers is the largest of their common divisors. We usually write it like this: GCD (a, b, c, …).

2nd We take the common factors raised to the smallest exponent.

My visual summary

Adding and subtracting two integers

The numbers have the same signs

• We add their absolute values.

• We use the same sign as the addends in the result.

4

Adding and subtracting integers

We have two options:

Do the operation step by step, in the order the numbers appear:

–

Group the positive numbers on one side, and the negative ones on the other. Then do the operation.

Multiplying and dividing integers

(+4) . (+3) = +12

(–5) . (–4) = +20

(+6) (–3) = –18

(–4) (+8) = –32

(+24) : (+4) = +6

(–24) : (–6) = +4

(+24) : (–4) = –6

(–24) : (+4) = –6

The numbers have different signs

• We subtract their absolute values.

• We use the sign of the number with the greatest absolute value in the result.

The sign of the number with the greatest absolute value.

Adding and subtracting with parentheses

+ sign in front of brackets

+(–3 + 8 – 2) = –3 + 8 – 2

Same sign

– sign in front of brackets

–(–3 + 8 – 2) = +3 – 8 + 2

Opposite sign

Combined operations

2nd Multiplication and division

1st Brackets

3rd Addition and subtraction

We calculate inside the brackets.

(–18) : (11 – 9 – 5) + 5 (6 – 8) = = (–18) : (–3) + 5 · (–2) = We do the divisions and multiplications.

= (+6) + (–10) = = 6 – 10 = –4

We do the additions and subtractions.

Powers of integers

A power is a multiplication of identical factors:

Powers of negative numbers

Even exponent

Odd exponent

positive result

negative result

Properties of powers

• The power of a product is equal to the product of the powers of the factors.

• The power of a quotient is equal to the quotient of the powers of the dividend and the divisor.

• To multiply two powers with the same base, we add up the exponents.

• To divide two powers with the same base, we subtract the exponents.

• To raise one power to another power, we multiply the exponents.

Roots of integers

A square root is the opposite of raising to the power of two (squaring).

Numbers that have an integer square root are called perfect squares.

Any positive number has two square roots, one negative and one positive.

49 = 7

400 = 20

49 is a perfect square.

400 is a perfect square.

Negative numbers do not have a square root. A root index can be greater than two. Does not exist.

Exercises and problems

DO YOU KNOW THE BASICS?

Numeral systems

1 Look at this number in two different numeral systems: Egyptian numeral system

Mayan numeral system

a) Explain the meaning of the symbols in each case.

b) Write the number before and the number after in both numeral systems.

2 Copy and complete.

a) 2 300 Th = … H b) 4 800 T = … Th

c) 2 HTh = … Th d) 700 Th = … TTh

3 Copy, calculate and complete.

a) 1 h 13 min 27 s 8 … s

b) 587 min 8 … h … min

c) 6 542 s 8 … h … min … s

Multiples and divisors

4 Answer. Justify your answer.

a) Is 132 a multiple of 11? Is 11 a divisor of 132?

b) Is 574 a multiple of 14? Is 27 a divisor of 1 542?

5 Calculate.

a) The first five multiples of 10.

b) The first five multiples of 13.

c) All the divisors of 23.

d) All the divisors of 32.

6 Think and answer.

a) The three biggest divisors of a number are 20, 30 and 60. What number is it?

b) The three smallest divisors of a number are 12, 24 and 36. What number is it?

Prime and composite numbers

7 Write:

a) The first ten prime numbers.

b) The biggest two-digit prime number and the smallest three-digit prime number.

8 Copy and complete. Decompose these numbers into prime factors.

1

9 Decompose into the maximum number of factors.

a) 378 b) 1 144 c) 1 872

Lowest common multiple and greatest common divisor

10 Calculate. Use mental arithmetic.

a) LCM (2, 3) b) LCM (6, 9)

c) LCM (4, 10) d) LCM (6, 10)

e) LCM (6, 12) f ) LCM (12, 18)

11 Calculate. Use mental arithmetic.

a) GCD (4, 8) b) GCD (6, 9)

c) GCD (10, 15) d) GCD (12, 16)

e) GCD (16, 24) f ) GCD (18, 24)

12 Calculate.

a) LCM (24, 36) b) GCD (24, 36)

c) LCM (28, 42) d) GCD (28, 42)

e) LCM (45, 75) f ) GCD (45, 75)

Integers

13 Order from smallest to biggest.

– 6, +8, –16, –3, +12, –7, +4, +15, –11

Adding and subtracting integers

14

Calculate.

a) 5 – 8 – 4 + 3 – 6 + 9

b) 10 – 11 + 7 – 13 + 15 – 6

c) 9 – 2 – 7 – 11 + 3 + 18 – 10

d) –7 – 15 + 8 + 10 – 9 – 6 + 11

15

Calculate.

a) 15 + (8 – 6) b) 11 – (2 + 8)

c) 6 + (2 – 8) – (1 + 7) d) (13 – 11) – (10 + 7) – (2 – 10)

Multiplying and dividing integers

16 Solve using the rule of signs.

a) (– 4) · (+7) b) (–21) : (+3)

c) (– 6) · (–8) d) (+30) : (+5)

e) (+10) · (+5) f ) (– 63) : (–9)

g) (–9) · (–5) h) (+112) : (–14)

17 Copy and complete.

a) (–3) · (…) = –15 b) (–28) : (…) = –4

c) (…) · (–4) = +32 d) (…) : (+5) = +10

e) (+20) · (…) = +60 f) (…) : (–7) = +8

Combined operations with integers

18 Calculate.

a) 5 – 4 · 3 b) 2

c) 4

5 – 6

e) 16 – 4

3 d) 2

7 + 2

19 Calculate

5 – 19 f ) 5

TRAINING AND PRACTICE

24 The numbers below are the first ten numbers in the binary system (it uses only the symbols 1 and 0).

0 - 1 - 10 - 11 - 100 - 101 - 110 - 111 - 1000 - 1001

Write the next ten numbers.

25 Copy these numbers and identify:

66 71 90 103 105

156 220 315 421 825

1 000 2 007 4 829 5 511 6 005

a) The multiples of 2. b) The multiples of 3.

c) The multiples of 5. d) The multiples of 11.

26 Write.

9 – 7

8 – 4

5

6 – 21 – 3

7 + 12

a) 7 · (6 – 4) b) (7 – 10) · 2

c) (–3) · (7 – 6) d) (10 – 4) · (–2)

e) 6 · (5 – 3) + 2 · (2 – 7) f) 5 · (–3 – 1) – 4 · (9 – 7)

Powers of integers

20 Calculate.

a) (–5)4 b) (+4)5 c) (– 6)3

d) (+7)3 e) (–8)2 f ) (–10)7

g) (+3)0 h) (–6)0 i) (–10)0

21 Express as the power of a single number.

a) 104 : 54 b) 127 : (– 4)7

b) (–9)6 : 36 d) 26 · 26

c) (– 4)5 · (–2)5 f ) 24 · (–5)4

22 Reduce to a single power.

a) x2 · x4 b) m4 · m3 c) x6 · x

d) m8 : m5 e) x3 : x f) m5 : m5

g) (x3)2 h) (m5)2 i) (x 2)2

Square roots of integers

23 Calculate when possible.

a) 49 b) 7 2 c) 49 –

d) 15 2 e) 225 f ) –225

g) 2 500 h) 50 2 i ) 2 500 –

a) The prime numbers between 50 and 60.

b) The prime numbers between 80 and 100.

c) The first three prime numbers greater than 100.

27 Calculate.

a) LCM (12, 15) b) LCM (24, 60)

c) LCM (48, 54) d) LCM (90, 150)

e) LCM (6, 10, 15) f ) LCM (8, 12, 18)

28 Calculate.

a) GCD (36, 45) b) GCD (48, 72)

c) GCD (105, 120) d) GCD (135, 180)

e) GCD (8, 12, 16) f ) GCD (45, 60, 105)

29 Write the coordinates of the vertices of this rectangle and draw another one like it, with its vertex M at point (1, 0).

30 Calculate.

a) 16 + [3 – 9 – (11 – 4)]

b) 8 – [(6 – 9) – (7 – 13)]

c) (6 – 15) – [1 – (1 – 5 – 4)]

d) (2 – 12 + 7) – [(4 – 10) – (5 – 15)]

e) [9 – (5 – 17)] – [11 – (6 – 13)]

Exercises and problems

31 Calculate.

a) (–2) · [(+3) · (–2)] b) [(+5) · (–3)] · (+2)

c) (+6) : [(–30) : (–15)] d) [(+40) : (– 4)] : (–5)

e) (–5) · [(–18) : (– 6)] f ) [(–8) · (+3)] : (– 4)

g) [(–21) : 7] · [8 : (– 4)] h) [6 · (–10)] : [(–5) · 6]

32 Calculate. Note that the result changes depending on the position of the brackets.

a) 17 – 6 · 2 b) (17 – 6) · 2

c) (–10) – 2 · (–3) d) [(–10) – 2] · (–3)

e) (–3) · (+5) + (–2) f ) (–3) · [(+5) + (–2)]

33 Calculate.

a) 5 · [11 – 4 · (11 – 7)]

b) (– 4) · [12 + 3 · (5 – 8)]

c) 6 · [18 + (– 4) · (9 – 4)] – 13

d) 4 – (–2)

[–8 – 3 · (5 – 7)]

e) 6 · (7 – 11) + (–5)

[5 · (8 – 2) – 4 · (9 – 4)]

34 Reduce to a single power.

a) (x 2)5 b) (m 4)3

c) [a 10 : a 6]2 d) (a · a 3)3

e) (x 5 : x 2) · x 4 f ) (x 6 · x 4) : x 7

35 Look. Reduce like in the example:

• () xx x · 63 23 2 == = x 3

a) () x 22 b) () m 32 c) () a 42

d) x 4 e) m 6 f ) a 8

THINK, APPLY, EXPRESS

36 Find a divisor of 427 with two digits.

37 A number less than 50 is a multiple of 6 and 7. What number is it?

38 We can put a group of 20 people into an exact number of rows and columns. For example, four rows and five columns.

But we can't do that with a group of 13 people. We can only put them in one row.

Find all the numbers between 150 and 170 that we can only put in one row.

39 A three-digit number is a multiple of 150 and a divisor of 2100. What number is it?

SOLVE SIMPLE PROBLEMS

Problems with natural numbers

40 Target 3.8. The World Health Organization recommends between 9 and 11 hours of sleep a night for 14-year olds. It recommends between 7 and 9 hours for 40-year-olds. What is the difference between the annual hours of sleep of a 14-year-old and a 40-year-old?

41 We have wooden cubes weighing 30 grams each and some glass marbles weighing 36 grams each. We put them on a weighing scale. The scale is balanced and there is a total of 15 cubes and marbles:

a) How much does each side weigh?

b) How many wooden cubes and how many marbles are there?

42 Two lorries transport identical fridges. The first has a cargo weighing 481 kilos, and the second, 555 kilos. How much does each fridge weigh? How many fridges are there in each lorry?

43 A roll of cable is longer than 150 m, but shorter than 200 m. We can divide it into sections of 15 m and 18 m with no remaining cable. How long is the cable?

44 A farmer divides a field into 15 m × 20 m rectangular pieces of land. There are nearly 50 pieces of land. What were the measurements of the field?

45 A cake shop sells 2 400 muffins and 2 640 mantecados in bags of the same number of units, but without mixing the products. There are between 10 and 15 units in each bag. How many muffins and mantecados are there in each bag?

46 Take Action. A lorry is 12 m long, 2.25 m wide and 2.40 m high. Design a box to transport products in the lorry. Find the optimal solution. You do not want any empty spaces.

12 m

Problems with integers

47 Draw a coordinate system and represent the points A (–2, 0) and B (4, 2). Draw all the squares that have their vertices at these points (there are three different squares). Finally, write the coordinates of the vertices of each of the squares.

48 If you write all the integers from –50 to +50, how many times does the digit 7 appear? And 5? And 3?

49 Problem solved

The sum of two integers is (–5) and their difference is (+19). What are the two numbers?

Let’s start with a very simple example.

With numbers 6 and 4:

If we subtract the difference from the sum of the two numbers, we get double the smaller number.

(a + b) – (a – b) = a + b – a + b = 2 b

Let’s solve the original problem.

– The sum is (–5) and the difference is (+19).

– The sum minus the difference is double the smaller number:

(–5) – (+19) = –5 – 19 = –24 (double the smaller number)

The smaller number is: (–24) : 2 = –12

The larger number is: –12 + 19 = 7

Now, check the answer.

50 The sum of two integers is 3 and their difference is 7. What are the two numbers?

51 The sum of two integers is –22, and the sum of their absolute values is 70. What are the two numbers?

THINK A LITTLE MORE

52 A local cake shop makes muffins every morning. They put them into bags of half a dozen muffins, with two left over.

If they put them into bags of 5, there are three muffins left over. If they put them into bags of 8 there are no muffins left over.

There are a little more than 40 bags. How many muffins are there?

53 The members of an athletics club have decided to give their coach a stopwatch that costs €130.

‘If three more people contributed, we would pay €3 less’ says the team captain.

How many people contribute to the stopwatch if each person pays an exact amount in Euros

54 My brother and I are going to buy a birthday present for our mum. After paying his part, my brother has €10 remaining. I borrow €5 from him to pay my part.

We have €85 between us. How much is the birthday present?

55 I have two accounts at the same bank. In the first one there are €200 more than in the second one. I transfer money from one to the other and now. I have €20 in each account.

How much money was in each account before?

Look at this graph:

You can also look at problem solved 49 again and think: What is the sum of both accounts and what is the difference?

1 Write.

a) The first four multiples of 17.

b) All the divisors of 72.

2 Find.

a) The first multiple of 17 after 1 000.

b) A two-digit number that is a divisor of 415.

3 Write the prime numbers between 20 and 40.

4 Indicate which of the following numbers are multiples of 2, 3, 5 and 10:

897 - 765 - 990 - 2 713 - 6 077 - 6 324 - 7 005

5 Think and calculate.

a) The largest number that is a divisor of 6, 12, 18 and 24.

b) The largest number that is a divisor of 3, 5, 7 and 11.

c) The smallest number that is a multiple of 3, 5, 7 and 11.

d) The smallest number that is a multiple of 6, 12, 18 and 24.

6 Copy and decompose the numbers 150 and 225 into prime factors in your notebook.

150 = 2 · · · 225 = 3 · · ·

7 Calculate.

a) GCD (150, 225) b) LCM (150, 225)

8 Calculate.

a) 6 – 11 + (9 – 13) b) 2 – (5 – 8)

c) (7 – 15) – (6 – 2) d) 5 – [2 – (3 – 2)]

9 Calculate.

a) 4 · 5 – 3 · (–2) + 5 · (–8) – 4 · (–3)

b) (10 – 3 · 6) – 2 · [5 + 3 · (4 – 7)]

c) 10 – 10 · [– 6 + 5 · (– 4 + 7 – 3)]

10 Reduce to a single power.

a) a 3 : b 3 b) a 5 : b 5 c) a 4 · a 2

d) x 6 . x 4 e) (x 3)3 f) (–5)7 : (–5)5

11 A clothes shop sells a number of T-shirts at the same price. The first day they sell T-shirts with a total value of €221, and on the second day they sell some more T-shirts, with a value of €272 . What is the price of one T-shirt? €?

12 A farmer has a field that is 100 m wide and 120 m long. She divides it into the largest square pieces of land possible. What do the sides of the pieces of land measure?

13 In a factory, there is a sound from a gas valve every 45 s, and a sound from a machine every 60 s. The valve and the machine make a sound simultaneously. When will the sounds next coincide again?

14 A tower is built using wooden cubes with 45 cm edges. Next to it, another tower is built with plastic cubes with 60 cm edges. At what height will both towers be the same size?

15 The sum of two integers is 4, and the sum of their absolute values is 16. What are the numbers?

16 Look at the square. A M B A' D C

a) Write the coordinates of the vertices, A, B, C, D, and of its centre, M.

b) Imagine you rotate it around M, so that A is on A l (4, 2). Write the coordinates of the new vertices, A l , B l , C l and D l .

Do you remember the storage problem at the beginning of the learning experience? You are now going to explore another similar problem. A company produces detergent and packs it in boxes of 40 × 30 × 25 cm. They want to optimise the space in the lorry and transport as many boxes as possible. You are going to think about how to position and stack the boxes.

Analyse how best to fill a space with objects of known dimensions 1

• First, let's think of a code to organise the data.

• What is the largest number of boxes you can pack in the truck? How do you put the boxes? How many boxes fit in each row, column and in total?

• If the boxes must be upright, with the 25 cm edge vertical, how many boxes can you pack now?

• What is the largest cube-shaped box that would completely fill the space?

• What other box models would completely occupy the space in the lorry?

Now,

• Find out the actual dimensions of a real lorry or van.

• Look on the Internet, or in a department store, for the sizes of packing boxes and choose one.

• Calculate the maximum number of boxes that you can pack in your vehicle.

• Explain how to put them, to optimise the space in the vehicle.