MUESTRA
José María Arias Cabezas Ildefonso Maza Sáez
1
E S O
A guide to your book With the book you are holding you will: ➜ Learn Maths using the 5-stage methodology: Engage, Explore, Explain, Elaborate and Evaluate, the same method used in the Spanish book. ➜ Consolidate and improve your English proficiency through activities where you will practise both written and oral comprehension and expression. The following information is given: 1. The content of the block and the 5 stages: ➜ Engage: a stimulating GeoGebra applet related to the core concepts covered in the block. ➜ Explore: an interactive and dynamic activity based on the applet that lets you explore your current knowledge and relate it to the concepts you will be studying. ➜ Elaborate: a section to investigate and apply the concepts. 2. A learning situation with two phases: ➜ Explore: searching for information on a concept to be worked on. ➜ Elaborate: a short writing activity to present the information obtained.
Main content Each unit’s double page contains the Explain stage where the core concepts are presented. This is supported by a solved exercise or problem to illustrate the mathematical concept. The exercise is solved with straightforward operations so the students can focus on the new content being covered. The final Elaborate section has exercises and problems to enhance and consolidate the acquired knowledge.
Learning experience A comprehensive learning experience will enable you to develop all the key competences and core concepts you have acquired, as well as enhance your teamwork skills. Additionally, you will be able to communicate the results in a creative manner, using different formats.
Self-evaluation The last step in the 5-stage methodology is Evaluate, which has two parts: ➜ A solved evaluation that serves as a review of the content covered. ➜ A proposed evaluation covering the block’s content.
Index 1 Natural numbers and divisibility 2 Integers 3 Fractions 4 Decimal numbers 5 Powers and square roots Learning experience. Self-evaluation
6 The metric system 7 Proportionality and percentages 8 Linear equations 9 Elements of a plane 10 Triangles
6 12 16 22 28 32 38 42 48 54 58
Learning experience. Self-evaluation
62
11 Polygons and circles 12 Perimeters and areas 13 Geometric bodies 14 Functions, tables, graphs and probability
68 74 80 84
MODULE
I
e
Core concepts 1 Natural numbers and divisibility 2 Integers 3 Fractions 4 Decimal numbers 5 Powers and square roots
e
e
Open the applet using the QR. 1 Observe the solved exercise and work out the proposed
exercise. 2 Explore with other fractions and draw conclusions.
Learning experience At the end of this module, you will find a global learning situation that will allow you to work in teams. You will need to apply all the core concepts you have acquired. In addition, you will be able to communicate the results in a creative manner, using different formats.
e
e
1 Find information on the short and long numerical scale and in
which countries each one is used. 2 Write a few sentences with the information you have obtained.
UNIT
1
Natural numbers and divisibility 1 How do we solve problems? What is the order of operations?
()
When an expression with natural numbers has multiple operations, they must be performed in this order:
· : + –
a) Brackets. b) Multiplication and division (from left to right). c) Addition and subtraction (from left to right).
6 · 4 : 2 = 24 : 2 = 12 12 : 4 · 3 = 3 · 3 = 9 12 : 6 : 2 = 2 : 2 = 1
a) 2 + 3 · 4
b) (2 + 3) · 4
c) 16 : 2 · 4
2 + 12
5·4
8·4
14
20
32
d) 36 : 4 – 3 · 2
9
– 6 3
What is the procedure for solving word problems? The procedure can be summarised in the following steps.
1
2
DATA Read the word problem carefully. Make a diagram if possible. Write down the data given.
4
6
UNIT 1
SOLUTION AND VERIFICATION Make sure the solution is reasonable. Write down the answers with the right units. Check that the conditions in the word problem are observed. Reread the problem and check the entire process.
3
QUESTION(S) Identify and write down what the problem is asking for.
APPROACH AND WORKING OUT Make an outline. Translate the word problem into an equation. Solve the equation carefully.
LET’S crack THIS! 1
A businesswoman buys 10 cases of soft drinks with 12 bottles each, and pays €6 for each case. If she sells each soft drink bottle for €2, how much will she earn by selling all the bottles?
Profit is the difference between
what she makes from the sale: €240
1. Data: • She buys 10 cases with 12 bottles in each case. • She pays €6 per case. • She sells each bottle for €2 2. Question: • What is her profit?
what she pays for the purchase: €60
3. Approach and working out: Profit = Sales value - purchase value Sales value = Number of bottles · price of each bottle Sales value = 10 · 12 · 2 = €240 Purchase value = Number of cases · price of each case Purchase value = 10 · 6 = €60 Profit = 240 – 60 = €180
4. Solution: Her profit is €180. This is a reasonable solution and is expressed in Euros. Verification: Purchase value + profit = Sales value 60 + 180 = €240
1. Perform the following operations:
5. A warehouse buys 500 cases of tomatoes. Each case
c) 15 + 5 · (20 + 15)
weighs 10 kg. The total price is €4 500 and transport costs €600. During transport some cases fall off the lorry and 500 kg of tomatoes are ruined. What must the warehouse charge per kilo to make a profit of €3 900?
d) 4 · (20 – 4) – (40 – 12) : 2
6. A stationery shop buys 500 pens for €6 each. Some
a) 5 + 4 · 8 – 25 : 5 b) 240 : 2 + 3 · 5
2. Ernesto has €230 saved in the bank. He gets €52
for his birthday and buys 3 books for €12 each. How much money does he have left?
3. A bookshop buys 40 books for €10 each. What
profit will it make if it sells all the books for €13 each? And if it sells only half the books for €15 each?
4. A hardware store buys 4 reels of cable with 200 m
on each reel at a price of €2 per metre. How much must it charge for one metre of cable in order to make €800 in profit?
of the pens are sold for €500, at €5 each. To make sure no money is lost, what must be the price charged for the rest of the pens?
7. It is estimated that an average 15-minute shower consumes 300 L of water. At Sonia’s house, they have decided to take 8-minute showers. In addition, with each shower they will collect 4 L of water for recycling while they wait for the hot water to come out. There are four family members in the house and they shower once a day. How many litres of water will they save during one year? Natural numbers and divisibility
7
2 How do we factor a number? What are prime and composite numbers? Multiples and divisors Multiples of 6 are: M(6) = {0, 6, 12, 18, 24…} The divisors of 6 are: D(6) = {1, 2, 3, 6}
A number a is a multiple of another number b if a can be divided by b exactly. A number b is the divisor of another number a if a can be divided by b exactly. You can also say that a is divisible by b or that b is a divisor or factor of a. Prime and composite numbers
The number 7 is prime. D(7) = {1, 7} The number 35 is composite. D(35) = {1, 5, 7, 35}
A natural number is prime if it has only two divisors, 1 and the number itself. A natural number is composite if it has more than two divisors. Divisibility criteria A number is divisible by 2 if its last digit is an even number: 0, 2, 4, 6, or 8
The number 1
The number 1 is neither prime nor composite because it has an inverse, which is 1. It has only one divisor, which is 1
The numbers 20, 42, 54, 76 and 98 are divisible by 2
A number is divisible by 3 if the sum of its digits is a multiple of 3 456 ⇒ 4 + 5 + 6 = 15 ⇒ The sum of the digits is 15, which is a multiple of 3
A number is divisible by 5 if it ends in 0 or 5 The numbers 20 and 145 are divisible by 5
A number is divisible by 9 if the sum of its digits is a multiple of 9 7425 ⇒ 7 + 4 + 2 + 5 = 18
⇒
The sum of the digits is 18, which is a multiple of 9
To find out if a number is divisible by 11, add the digits in the even numbered positions and subtract the sum of the digits in odd numbered positions. The number is divisible by 11 if the result is a multiple of 11 18372915 Add the digits in the even-numbered positions: 8 + 7 + 9 + 5 = 29 Add the digits in the odd-numbered positions: 1 + 3 + 2 + 1 = 7 29 – 7 = 22, which is a multiple of 11
8
UNIT 1
What is the procedure for factorization? Factorial decomposition or factorisation of a number consists of expressing the number as the product of its prime factors with the corresponding exponents. Simple examples: These factorisations can be worked out mentally. 4 = 22 8 = 23
6=2·3 12 = 22 · 3
Procedure for factoring large numbers a) Write the number and draw a vertical line to the right of it. b) If the number ends in zero, it is divisible by 10 = 2 · 5. On the right side of
the vertical line, write 2 · 5 with an exponent equal to the number of zeros in the number.
c) Continue dividing each resulting quotient by the smallest prime number
(2, 3, 5, etc.) that is a divisor. Do this as many times as possible.
d) The operation is finished when the quotient is 1
2
Remember
When you see the you should do the operation mentally.
Factor the number 120 Quotients 120 0
10 12 0
2 6 0
2 3 0
3 1
120 12 6 3 1
Prime factors
Factorisation
2·5 2 2 3
120 = 23 · 3 · 5
3
Factor the number 36 000 36 000 23 · 53 36 2 18 2 9 3 3 3 1 36 000 = 25 · 32 · 53
8. Which numbers are prime? Which numbers are
composite? 7, 12, 13, 25, 31, 43
11. Factor the numbers below into primes. Do it mentally for section a.
9. Of the numbers 22, 30, 65, 72, 81 and 891, which
a) 4, 6, 9, 12 and 15
ones are:
a) divisible by 2
b) divisible by 3
c) divisible by 5
d) divisible by 6
d) divisible by 9
f) divisible by 11
10. Calculate x in the number 35x so that it is divisi-
ble by:
a) 2
b) 2 and 5
c) 3
d) 6
b) 180, 200, 475, 540 and 625 12. Make a Sieve of Eratosthenes. Copy the natural
numbers from 2 to 100. Cross out multiples of 2, except for 2 itself, starting with 22 = 4. Then cross out multiples of 3, except 3 itself, starting with 33 = 9. Continue to apply the same process with 5 and 7. The numbers that remain are primes less than 100
Natural numbers and divisibility
9
3 What is the Greatest Common Divisor
and the Least Common Multiple?
How do we calculate the Greatest Common Divisor? The greatest common divisor, or GCD of two or more numbers a, b, c, d, e, etc. is the largest of their common divisors. It is written as:
Notation
∩ is the symbol for ‘intersection�. It means elements in both of two different sets.
GCD(a, b, c, d, e, etc.) 4
Calculate the greatest common divisor of 12 and 18
The divisors for 12 are: {1, 2, 3, 4, 6, 12} D(18)
D(12) 4
1
2
9
12
3
6
18
The divisors for 18 are: {1, 2, 3, 6, 9, 18} The common divisors are: D(12) ∩ D(18) = {1, 2, 3, 6} The greatest common divisor is 6. It is written as GCD(12, 18) = 6 The two numbers a and b are coprime numbers if the(a, b) = 1 5
Determine if the following pairs of numbers are coprime:
a) 8 and 15
b) 9 and 12
a) GCD(8, 15) = 1, therefore 8 and 15 are coprime. b) GCD(9, 12) = 3, therefore 9 and 12 are not coprime.
Method for calculating the greatest common divisor in simple cases In these cases, you can determine the GCD mentally GCD(2, 6) = 2 GCD(6, 8) = 2
GCD(6, 9, 15) = 3 GCD(15, 20, 25) = 5
Method for calculating the greatest common divisor for large numbers a) Factor the numbers into primes. b) Find all common prime factors with the lowest exponent and multiply
them.
6
Calculate the GCD for 80 and 140 80 8 4 2 1
10
UNIT 1
2·5 2 2 2
140 14 7 1
2·5 2 7
80 = 24 · 5 ⎫ ⇒ 140 = 22 · 5 · 7 ⎬⎭ GCD(80, 140) = 22 · 5 = 20
How do we calculate the Least Common Multiple? The least common multiple, or LCM, of two or more numbers a, b, c, d, etc. is the smallest of their common multiples that is not zero. It is written as: LCM(a, b, c, d, etc.) 7
Calculate the least common multiple of 4 and 6
M(6)
M(4)
The multiples of 4 are M (4) = {0, 4, 8, 12, 16, 20, 24, 28, 32, 36…} The multiples of 6 are M(6) = {0, 6, 12, 18, 24, 30, 36, 42…} The common multiples are M(4) ∩(6) = {0, 12, 24, 36…} Of these common multiples, the lowest that is not zero is 12
4, 8, 16, 20, 28, 32…
0 24
…
12 36
6, 18, 30, 42…
This is written as: LCM(4, 6) = 12 Method for calculating the least common multiple for large numbers a) Factor the numbers into primes. b) Find the common and uncommon prime factors with the highest
exponent and multiply them.
8
Calculate the least common multiple of 45 and 60 45 15 5 1
3 3 5
60 6 3 1
2·5 2 3
45 = 32 · 5 ⎫ ⎬ ⇒ LCM(45, 60) = 22 · 32 · 5 = 180 2 60 = 2 · 3 · 5 ⎭
13. Find:
16. Calculate mentally:
a) GCD(250, 60)
a) LCM(20, 40)
b) GCD(75, 105)
b) LCM(6, 15)
c) GCD(135, 225)
c) LCM(4, 9)
d) GCD(200, 250) 14. Calculate mentally: a) GCD(4, 6, 8) b) GCD(20, 10, 4) c) GCD(20, 35, 45) d) GCD(98, 126, 140) 15. A farm has 264 hens and 450 roosters. They have
to be transported in cages without being mixed. The cages must hold as many animals as possible and each cage must have the same number of animals. How many animals go in each cage?
Method for calculating the least common multiple in simple cases
When the numbers are simple, you can find the LCM mentally. LCM(2, 6) = 6 LCM(2, 5) = 10 LCM (6, 9) = 18 LCM (3, 4, 6) = 12
d) LCM(14, 21) 17. Find: a) LCM(64, 80) b) LCM(140, 220) c) LCM(135, 225) d) LCM(200, 250) 18. Ana takes the paper rubbish out to the waste bin
every 12 days. Sonia does the same every 15 days. If they meet on a certain day, how many days later will they meet again? Natural numbers and divisibility
11
UNIT
2
Integers 1 What are integers and their uses? What are the uses of positive and negative numbers? There are situations that cannot be expressed mathematically using only natural numbers. Sometimes, when you indicate an amount, you need to indicate a direction relative to a point of origin. In such situations, you use negative numbers. Some situations that require positive and negative numbers to express the amounts are: • Position: positive for above sea level or negative for below sea level. • Money: positive for having money or negative for owing money. • Temperature: positive for above zero or negative for below zero. • Time: positive for after 1 AD (Anno Domini) or negative for before 1 BC (Before Christ). Situation
Position
Money
Temperature
Time
ℕ
0
1
2
4
5
6 … …
–2 –4
A drone flies at a height of 25 m
25
A diver dives to a depth of 10 m
–10
You have €1 200 in the bank.
1 200
You owe €15
–15
It is – 8 °C in the mountains.
–8
It is 38 °C on the beach.
38
Cleopatra was born in 69 AD
– 69
Michael Stifel was born in 1487
1487
ℤ
–1
3
Representation
–3
What are integers? The set of all What are integers? is comprised of the set of natural numbers ℕ = {0, 1, 2, 3…} and negative numbers {–1, –2, –3…} The set of integers is represented by the letter ℤ: ℤ = {… –4, –3, –2, –1, 0, 1, 2, 3, 4…} = {0, ±1, ±2, ±3, ±4…}
12
UNIT 2
What is absolute value? The absolute value of an integer is the number without a sign. You can understand this through geometry. The absolute value of number a is the length of a segment that has zero as its origin and ends at number a To represent the absolute value of number a, write the number between two vertical lines |a|. It is read as ‘the absolute value of a’ 1
Calculate the absolute value of −5 and 4 ∙4∙ = 4
∙– 5∙ = 5 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
8
9
What is the opposite of a number? The opposite of an integer a is another integer −a that has the same value, but the opposite sign. Notice that the opposite of a number is the same distance from zero as the number, but in the opposite direction; in other words, it is symmetrical in relation to zero. a) The opposite of 4 is −4
–4
0
4
b) The opposite of −3 is 3
–3
0
3
1. Assign a positive or negative number to each of the following situations.
4. Write two numbers that have the same absolute value.
a) You are in the second basement floor of a building.
5. Write the statements below mathematically.
b) The temperature of some water is 7 °C
a) I had €120 and I paid €20
c) Pedro owes Luis €3
b) I went up 4 floors and then I went down floors.
d) I have saved €12
c) My father gave me €5 and I spent €6
2. Write five integers that are not natural numbers.
d) It was 2 °C and the temperature decreased by 5 °C
3. The absolute value of a number is 6. What number
6. Calculate the absolute value of the numbers:
can it be?
–6
6
0
3
–2 Integers
13
2 How do we add and subtract integers? Adding two integers Two numbers with the same sign To add two numbers with the same sign, add their absolute values and use the same sign as the numbers. 2
Calculate: a) 4 + 6 b) – 4 + (– 6) a) 4 + 6 = 10 → I have 4 and 6, in total I have 10 b) – 4 + (– 6) = – 10 → I lose 4 and 6, in total I lose 10 4
– 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 –1 0
1
2
6 3
4
5
6
7
8
9 10
Two numbers with different signs To add two numbers with different signs, use the sign of the number with the greater absolute value and subtract the number with the lesser absolute value. 3
Calculate: a) 4 + (– 6) b) – 4 + 6
Multiplication (+) · (+) = +
Positive multiplied by positive = positive
(–) · (–) = +
Negative multiplied by negative = positive
(+) · (–) = –
Positive multiplied by negative = negative
(–) · (+) = –
Negative multiplied by positive = negative
a) 4 + (– 6) = – 2 → I have 4 and I lose 6. I lose a total of 2 b) – 4 + 6 = 2 → I lose 4 and I get 6. I have a total of 2
Subtracting two integers To subtract two integers add the first to the opposite of the second. a) 6 – 4 = 6 + (– 4) = 2 c) 6 – (– 4) = 6 + 4 = 10
b) 4 – 6 = – 2 d) – 4 – 6 = –10
Division (+) : (+) = +
Positive divided by positive = positive
(–) : (–) = +
Negative divided by negative = positive
The sign rule
(+) : (–) = –
Positive divided by negative = negative
Use the sign rule to multiply or divide integers.
(–) : (+) = –
Negative divided by positive = negative
If you multiply or divide integers with the same sign, the result is positive. If they have different signs, the result is negative.
14
UNIT 2
Multiplying and dividing integers
Multiplication and division To multiply or divide two integers, first determine the sign of the result. Then multiply them or divide them as if they were natural numbers. Multiplication
Division
Example
Example
4 · 5 = 20
20 : 4 = 5
– 4 · (– 3) = 12
– 12 : (– 3) = 4
8 · (– 3) = – 24
24 : (– 3) = – 8
– 7 · 2 = – 14
– 14 : 2 = – 7
What is the order of operations? ()
When an expression with integers has multiple operations, they must be performed in this order:
· :
a) Parentheses. b) Multiplication and division (from left to right).
+ –
c) Addition and subtraction (from left to right).
3 · (7 – 2) + 4 = 3 · 5 + 4 = 15 + 4 = 19 Brackets [ ] 4
2 – [5 – (6 · 3 – 4)]
Calculate:
a) 5 · (7 – 3) + 8
b) 3 + (–12 + 4) : 2
c) 18 : 6 · 2
d) 24 : 4 : 3
a) 5 · (7 – 3) + 8
b) 3 + (–12 + 4) : 2
c) 18 : 6 · 2
d) 24 : 4 : 3
a) a) a) a) a)
5 · 4 + 8
3 + (– 8) : 2
3·2
6:3
20 + 8
3 – 4
6
2
28
–1
2 – [5 – (18 – 4)] 2 – (5 – 14) 2 – (– 9) 2+9 11
7. Perform these operations mentally. a) 7 + 5
b) – 3 + (– 6)
9. Perform these operations. a) 2 · 6 – 10 + 5 + 15 : 5
c) – 8 + 12
d) 9 + (– 3)
b) – 2 · 6 + 3 · 5 – 12 : 2
8. Perform these operations mentally. a) 6 · 5 b) – 3 · (– 7)
10. Perform these operations. a) 15 – (8 – 5 + 9 + 2)
c) 15 : (– 3)
b) 25 + 40 : 2 – [5 – (8 – 9)]
d) – 36 : 12
Integers
15
UNIT
3
Fractions 1 How do we work with fractions? What is a fraction?
A fraction is the quotient of two integers; the divisor cannot be zero. 4 = 0,8 5 4 ab/c 5 = 4 5 ab/c 0,8 ab/c 4 5
The numerator is how many parts of the whole you have. The denominator represents how many equal parts the whole is divided into. 1
Show the fraction 3/4 of a square and 7/4 with circles.
Numerator ➞ 3 Denominator ➞ 4
Numerator ➞ 7 Denominator ➞ 4
Fractions as operators 500 : 5 = 100 100 · 3 = 300
Fractions also act as operators or functions on a given amount. To calculate the fraction of a given amount, divide the amount by the denominator and multiply the result by the numerator. 2
How much is 3/5 of a 500 gr bag of almonds? 3 · 500 = 500 : 5 · 3 = 300 gr of almonds. 5
What are equivalent fractions? Two fractions are equivalent if they express the same amount. The best way to test if two fractions are equivalent is to apply the cross multiplication rule, which states: Two fractions are equivalent if the products from a cross multiplication are equal. →→
3 = 5
→→
6 ⇒ 3 · 10 = 6 · 5, that it is to say, 30 = 30 10
Reducing fractions to the least common denominator Use this procedure to reduce fractions to their least common denominator: a) The least common denominator is the least common multiple (LCM) of
the denominators. b) The new numerators are obtained by dividing the LCM by the original denominators and then multiplying by the corresponding original numerator. 16
UNIT 3
3
Find the least common denominator of 3/4 and 5/6 LCM(4, 6) = 12
3 = 12 : 4 · 3 = 9 4 12 12
5 = 12 : 6 · 5 = 10 6 12 12
Comparing and ordering fractions When comparing fractions there are three possible cases. a) If they have the same denominator, the smallest fraction is the one with the smallest numerator.
3< 4 5 5
b) If they have the same numerator, the smallest one is the one with the
2 < 2 7 5
largest denominator.
c) If they have different denominators and numerators, you must convert
them to their least common denominator, as explained before. The smallest one is the fraction with the smallest numerator.
4
Put 2/3 and 3/4 in order, from smallest to largest. 2 = 12 : 3 · 2 = 8 and 3 = 12 : 4 · 3 = 9 3 12 12 4 12 12 since 8 < 9 therefore 2 < 3 12 12 3 4
LCM(3, 4) = 12
What are irreducible fractions? A fraction is irreducible if it cannot be simplified, in other words, the numerator and denominator are coprime. Use this procedure to calculate an irreducible fraction. a) Find the GCD of the numerator and the denominator. b) Divide the numerator and the denominator by their GCD.
Whenever possible, simplify the fraction and make it irreducible.
1. Draw a square and shade 3/4 of it.
Find the irreducible fraction equivalent to 12/18: 12 = 12 : 6 = 2 18 18 : 6 3 GCD(12, 18) = 6
Using a calculator: 12 ab/c 18 = 2 3
6. Reduce to the least common denominator.
3, 5, 7 4 6 8
2. Show 7/5 using circles. 3. Calculate:
5
7. Simplify these fractions until you reach an
a) 2/3 of 18 b) 4/7 of 35 4. You have a kilo of oranges and use 3/5 to make
marmalade. How much do you have left?
5. Which fractions are equal to one another?
4 , 8 , 2 , 4 , 10 6 10 3 5 15
irreducible fraction. a) 6 b) 10 8 15
c) 12
18
d) 18
24
8. Ana, Maria and Pedro each buy a soft drink. After
10 minutes, Ana still has 1/2 of hers, Maria has 3/4, and Pedro has 1/3. Put the three friends in order according to how much of their drinks they have left. Fractions
17
2 How do we add and subtract fractions? Adding and subtracting fractions Fractions with same denominator The sum or difference of fractions with the same denominator is another fraction that has: a) Numerator: the sum or difference of the numerators. b) Denominator: the same as the original fractions.
The last step is to find the irreducible fraction.
6
Calculate
5 1 7 4 + – + 9 9 9 9
5 + 1 – 7 + 4 = 5 + 1 – 7 + 4 = 10 – 7 = 3 = 3 : 3 = 1 9 9 9 9 9 9 9 9:3 3 GCD(3, 9) = 3
+
–
5 9
1 9
+ 7 9
= 4 9
3= 1 9 3
Fractions with different denominators The sum or difference of fractions with different denominators is another fraction that has: a) Denominator: the least common multiple (LCM) of the denominators. b) Numerator: the sum or difference of the new numerators obtained by
dividing the LCM of the denominators by the original denominators and then multiplying by the corresponding original numerator.
The last step is to find the irreducible fraction.
The opposite fraction 4 4 is – of 7 7 Check: =
18
( )
4 4 + – = 7 7
4–4 0 = =0 7 7
UNIT 3
7 5 3 – + = 3 2 4
28 – 30 + 9 37 – 30 7 12 : 3 · 7 – 12 : 2 · 5 + 12 : 4 · 3 = = = 12 12 12 12
lcm(3, 2, 4) = 12
What are opposite fractions? The opposite fraction of a fraction is the result obtained when you change its sign. The sum of two opposite fractions is zero.
Adding and subtracting fractions and integers To add or subtract fractions and integers, the integers are treated as fractions with the denominator 1. The last step is to find the irreducible fraction. a) 3 + 2 = 3 + 2 = 5 · 3 + 2 = 15 + 2 = 17 5 1 5 5 5 5 3 + 2 ab/c 5 = 17 5
2 3 2 = + 5 1 5 3 2 3 2– = – 4 1 4
3+
b) 2 – 3 = 2 – 3 = 4 · 2 – 3 = 8 – 3 = 5 4 1 4 4 4 4 2 – 3 ab/c 4 = c)
5 4
3 + 5 – 9 = 12 · 3 + 12 : 6 · 5 – 12 : 4 · 9 = 36 + 10 – 27 = 46 – 27 = 19 6 4 12 12 12 12 lcm(6, 4) = 12
3 + 5 ab/c 6 – 9 ab/c 4 =
19 12
LET’S crack THIS! 7
At Ana’s house they collect rainwater and reuse it for watering plants, cleaning the yard, etc. They have managed to reuse 3/10 of the total water they use. They have made changes to their system and have collected an additional 1/2 of the total water used. What fraction of the total water used in the house is not yet reused? 1. Data • They reuse 3/10 of total water used. • They improve the system and reuse an additional 1/2 of total water. 2. Question: • How much water is not yet reused?
3. Approach and working out: • They reuse: 3 + 1 + 3 + 5 + 8 = 4 10 2 10 10 5 • They do not yet reuse: 1 – 4 = 1 5 5
4. Solution: They do not yet reuse 1/5 of the total water used.
9. Calculate mentally.
13. Perform the operations mentally.
a) 1 + 1
a) 3 + 5
b) 1 – 1 2 2 4 10. Calculate the following operations mentally. a) 2 – 4 + 7 b) 3 + 2 – 6 3 3 3 5 5 5 11. Perform the following operations. a) 1 – 5 + 7 b) 5 + 1 – 8 4 8 6 2 6 3 12. Perform the following operations. a) 11 – 5 – 3 b) 13 + 7 – 11 12 18 4 5 10 20
b) 5 – 4 4 6 14. Perform the following operations. a) 16 – 3 + 7 b) 7 – 5 – 4 + 9 5 10 6 4 2 15. Calculate the opposite fractions for the fractions below and check your answers. a) 2 b) – 4 5 3 16. An empty 1-litre bottle is filled with water up to 2/3 and then 1/4 more is added. How much more is needed to fill the bottle? Fractions
19
3 How to divide and multiply fractions? Multiplying fractions The product of two fractions is another fraction that has: a) Numerator: the product of the numerators. b) Denominator: the product of the denominators.
The last step is to find the irreducible fraction. 8
Sara has used 2/3 of a plot of land to plant vegetables. She wants to grow tomatoes on 4/5 of the planted area. What fraction of the total plot will she use to grow tomatoes? 2 3
4 5
2 · 4 = 2·4 = 8 3 5 3 · 5 15
2 ab/c 3 × 4 ab/c 5 =
8 15
The product of an integer and a fraction is another fraction that has: 4 · 3 = 4 · 3 = 4 · 3 = 12 5 5 1 5 5 4 × 3 ab/c 5 = 12 5
a) Numerator: the product of the integer and the numerator of the fraction. b) Denominator: the denominator of the fraction.
Reciprocal of a fraction The reciprocal fraction of 4 is 5 5 4 because 4 · 5 = 4 · 5 = 20 = 1 5 4 5 · 4 20 4 ab/c 5 = x –1 = 5 4
KCF
Apply the rule KCF = Keep, Change, Flip Keep 3/4 Change divide for multiply Flip 5/6 to 6/5
You create the reciprocal of a fraction by inverting the positions of numerator and the denominator. The sign does not change. The product of two reciprocal fractions is one. Zero does not have a reciprocal fraction.
Dividing fractions To divide two fractions, multiply the first by the reciprocal of the second; this is the KFC method (keep, change, and flip). Keep the first fraction as it is, change the division sign to multiplication sign, and flip the second fraction to its reciprocal. 3 3 5 3 6 9 : = · = 4 6 4 5 10 2 KCF 7:
Reminder
When working with fractions, at each step we must simplify and find the irreducible fraction.
20
UNIT 3
3 7 3 7 4 28 = : = · = 4 1 4 1 3 3
3 ab/c 4 ÷ 5 ab/c 6 = 9 10
7 ÷ 3 ab/c 4 = 28 3
KCF
2 2 7 2 1 2 :7= : = · = 3 3 1 3 7 21 KCF
2 ab/c 3 ÷ 7 = 2 21
Order of operations When an expression with fractions has multiple operations, they must be performed in this order: a) Brackets.
()
b) Multiplication and division (from left to right).
· :
c) Addition and subtraction (from left to right).
+ –
3 1 3 5 5 3 6 5 9 5 20 – 9 5 11 5 11 2– : · = 2– · · = 2– · = · = · = 4 6 7 4 5 7 10 7 10 7 10 7 14 2 2 KCF
(
) (
) (
)
( 2 – 3 ab/c 4 ÷ 5 ab/c 6 ) × 5 ab/c 7 = 11 14
LET’S crack THIS! 9
We travel 2/5 of a route and then 1/2 of the remaining route. What fraction of the route have we travelled so far? 1. Data • We travel 2/5 and then 1/2 of the remaining route. 2. Question: • What fraction of the total route have we travelled? 3. Approach and working out: 2 + 1 · 3 = 2 + 3 = 4 + 3= 7 5 2 5 5 10 10 10 4. Solution: We have travelled 7/10 of the total route.
17. Solve the following multiplication problems. a) 4 · 5 3 7 d) 6 · 7 8
b) 8 · 15 5 14 7 e) · 10 2
c) 2 · 4 · 6 3 5 7 f) 4 · (–12) 3
18. Calculate the reciprocals for each fraction and
check your answer.
a) 4 b) – 5 c) 2 d) – 1 7 3 6 19. Solve the following division problems. a) 2 : 7
b) 6 : 8 c) – 3 : 5 5 8 5 9 4 6 20. A car has a 60-kWh battery. It uses 2/3 of its capacity on a 300 km drive. If one kWh costs €0.08, how much did the trip cost?
21. Perform the operations. a) 7 : 3
b) 3 : 6 c) – 6 : (–9) 5 4 5 22. Perform the combined operations. a) 3 · 5 + 7 : 14
4 6
8
5
( 4 5) 2
c) 4 – 3 · 6 : 5
(4 8) 2 ( 34 : 65 – 2) · 49
b) 6 · 7 – 3 – 5
5
d)
23. You buy 200 kg of almonds at €5 per kilo. You
sell bags containing 1/4 kilo for €4 each. How much profit do you make?
24. A perfume shop buys a 40 L container of cologne for
€60 and sells it in 1/5 L bottles. Each empty bottle costs 1 € and, when full of cologne, is sold for €5. How much profit does the shop make? Fractions
21
UNIT
4
Decimal numbers
1 How do we work with decimal numbers? Adding and subtracting decimal numbers Procedure: a) Place the numbers one on top of another so that the decimal points are
aligned, in other words, in the same position for all the numbers.
b) Add or subtract as if they were natural numbers.
6.5 0.84 + 32.53 39.87
c) In the result, put the decimal point in the same position as the decimal
points in the problem. 1
Lola buys a carton of milk for €6.50, a packet of crisps for €0.84 and a bottle of olive oil for €32.53. How much does she spend in total? She spends 6.5 + 0.84 + 32.53 = €39.87
2
Pedro has €83.74 and spends €7.28. How much does he have left? He has 83.74 – 7.28 = €76.46
6.5 + 0.84 + 32.53 = 39.87
83.74 – 7.28 76.46 83.74 − 7.28 = 76.46
675.34 – 483.58 191.76 675.34 − 483.58 = 191.76
a) If the minuend has fewer digits to the right of the decimal point than the
subtrahend, you can add zeros to the right of the minuend or you subtract from 10 without adding the zeros.
3
A self-employed person bills €675.34 in VAT (value-added tax) in a quarter and has €483.58 in tax deductions. How much VAT does the person have to pay to the tax authorities? The amount of tax to be paid is 675.34 – 483.58 = €191.76
Multiplying decimal numbers Procedure: a) Line up the numbers on the right; do not align the decimal points.
27.39 × 4.6 16 434 109 56 1 2 5. 9 9 4 27.39 × 4.6 = 125.994 22
UNIT 4
b) Multiply as if they were natural numbers. c) Place the decimal point in the answer. Start at the right and move a number
of places equal to the total number of decimal places in the numbers multiplied.
d) If there are not enough decimal numbers in a product to equal the sum of
decimal places of numbers multiplied, place as many zeros as needed before the significant figures (i.e. to the right of the decimal point).
4
0.35 × 0.07 0. 0 2 4 5
Calculate 7 % of 0.35 by multiplying 0.35 · 0.07 7 % of 0.35 is 0.35 · 0.07 = 0.0245
0.35 × 0.07 = 0.0245
Remember that if there is a zero in the multiplier, it is not multiplied and the partial product moves one position to the left. 5
47.5 3.06 2 850 1425 1 4 5. 3 5 0 ×
You buy 47.5 litres of olive oil at €3.06 per litre. How much do you pay? You pay 47.5 · 3.06 = €145.35
47.5 × 3.06 = 145.350
Multiplying decimal numbers by multiples of 10 To multiply a decimal number by a digit followed by zeros, (in other words, by a multiple of ten), write the result and move the decimal point to the right the number of places equal to the number of zeros in the digit. Sometimes when you need to move the decimal point there are not enough decimal places. In that case, add as many zeros as necessary to the end of the number.
3.476 · 100 = 347.6 7.2 · 1 000 = 7 200
Multiplying decimal numbers by another decimal number starting with 0 To multiply a decimal number by another decimal number starting with a zero, write the result and move the decimal point to the left the number of places equal to the total number of decimal places in the numbers multiplied. If there are not enough digits, place as many zeros to the left as necessary.
485.7 · 0.01 = 4.857 5.23 · 0.001 = 0.00523
1. Add the decimal numbers.
5. Multiply the numbers mentally.
a) 45.23 + 7.842
b) 136.25 + 7.8 + 38.967
a) 8.19 · 0.01
c) 45.3 + 802.762
d) 0.0034 + 7.23 + 99.1
b) 234.56 · 0.001
2. Subtract the decimal numbers. a) 83.27 – 67.15
b) 8.5 – 3.47
c) 823.7 – 97.234
d) 2.567 – 0.58
3. Multiply the decimal numbers. a) 5.23 · 7.5
b) 23.9 · 8.4
c) 34.89 · 20.5
d) 0.00678 · 0.05
4. Multiply the decimal numbers mentally. a) 7.45 · 100
b) 0.056 · 10
c) 456.783 · 10 000
d) 0.00876 · 1 000
c) 659.23 · 0.0001 d) 0.023 · 0.1 6. To make a paella, you need the following ingredients: 0.4 kg of rice, 0.25 kg of calamari, 0.35 kg of clams and 0.27 kg of prawns. What is the total weight of all the ingredients? 7. Find the perimeter of a rectangle with sides measuring 5.7 m and 6.8 m 8. You buy 100 bags of crisps. Each bag weighs 0.25 kg. What is the total weight of the bags of crisps? Decimal numbers
23
2
How do we divide decimal numbers? Dividing integers when the quotient is a decimal number Procedure:
38 30 0 20 0.06
7 5. 4 2
a) Divide the integers. b) Put a decimal point in the quotient. c) Bring a zero down.
Q = 5.42; R = 0.06
d) Continue dividing.
38 ÷ 7 = 5.42857
e) Dividing decimal numbers
Dividing decimal numbers There are two possibilities: Only the dividend has decimals Procedure: a) Start dividing as if the numbers were natural numbers. b) When you arrive at the decimal point in the dividend, place a decimal
point in the quotient.
c) Continue dividing. 6
Calculate 9.75 : 4 9.75 1 7 0 15 0.03
4 2. 4 3
Q = 2.43; R = 0.03 9.75 ÷ 4 = 2.4375
The divisor has decimal numbers Procedure: a) Remove the decimals from the divisor. Multiply both, the dividend and
the divisor by the unit followed by as many zeros as there are decimal places in the divisor.
b) Perform the resulting division. Number of decimals in the remainder
The number of decimals in the remainder is found by lowering the initial decimal point of the dividend to the remainder.
24
UNIT 4
7
Calculate: a) 94.71 : 3.8
b) 68.3 : 5.47
a) 9 4. 7. 1
b) 6 8. 3 0
3. 8 2 4. 9
18 7 3 5 1 0. 0 9 Q = 24.9; R = 0.09
5. 4 7 1 2. 4
13 6 0 2 6 60 0 . 4 7 2 Q = 12.4; R = 0.472
Dividing decimal numbers by multiples of 10 To divide decimal numbers by a digit followed by zeros (in other words, by a multiple of 10), move the decimal point to the left as many places as the number of zeros after the digit. If there are not enough digits, put as many zeros to the left as necessary. a) 742.5 : 100 = 7.425
b) 8.2 : 1 000 = 0.0082
Dividing a decimal number by another decimal number starting with 0 To divide decimal numbers by a decimal number starting with a zero, move the decimal point to the right as many places as the number of zeros in the decimal number starting with a zero. If there are not enough digits, put as many zeros to the right as necessary.
What is the order of operations?
a) 8.965 : 0.01 = 896.5 b) 73.6 : 0.001 = 73 600
() · : + –
When there are several operations in a problem with decimal numbers, follow this order: (3.13 – 0.75) : 1.7 + 2.85 = = 2.38 : 1.7 + 2.85 = = 1.4 + 2.85 = 4.25
a) Parentheses. b) Multiplication and division (from left to right). c) Addition and subtraction (from left to right).
9. Divide to two decimal places.
14. Solve the following combined operations.
a) 31 : 8 c) 345 : 11
a) 4.5 + 2.5 · 7.8
b) 13 : 7 d) 5 : 13
10. Divide to two decimal placesa) a) 83.5 : 9 c) 5.93 : 17
b) 634.83 : 23 d) 587.4 : 47
11. Divide to two decimal placesa) a) 847.23 : 6.5 c) 0.485 : 3.25
b) 7.2 : 0.03 d) 8.345 : 3.47
12. Divide mentallya) a) 738.3 : 100 c) 76.34 : 10 000
b) 0.044 : 10 d) 34.2 : 1 000
13. Divide mentallya) a) 7.23 : 0.01 c) 3.2 : 0.0001
b) 0.0056 : 0.001 d) 678.5 : 0.1
b) 36.25 : 6.25 – 2.44 c) 3.2 · (56.3 + 6.98) d) (45.6 – 0.48) : 1.2 15. A warehouse manager buys 1 200 litres of a
beverage and puts it in 1.5 L bottles. How many bottles are filled?
16. A car with 35 litres of petrol travels 538 km. If a
litre of petrol costs €1.45, how much does it cost to travel one kilometre?
17. For a graduation party, 28 students buy 30 litres
of soft drinks at €1.20 per litre, 12.5 kg of chips at €5.70 per kilo and party decorations for €8.50. How much does each student have to pay?
Decimal numbers
25
3 How do we round decimal numbers and solve
problems?
How do we round decimal numbers? Rounding means making a number simpler by removing part of it. To round a number, if the number you eliminate is: a) 0, 1, 2, 3, or 4, then the number you want to round does not change. b) 5, 6, 7, 8, or 9, then you add one to the number you want to round. 8
Round the numbers below to two decimal places: a) 6.82465 b) 2.83593 c) 5.42723 d) 48.59642
The Guggenheim Museum in Bilbao receives an estimated 1 200 000 visitors a year.
a) 6.82465 = 6.82
b) 2.83593 = 2.84
c) 5.42723 = 5.43
d) 48.59642 = 48.6
How do we round decimals using a calculator? To round a number with a calculator, use: MODE (Fix) 1 And then enter the number of decimal places you want to round to. For example, to round to two decimal places, use: MODE (Fix) 1 (Fix 0 ∼ 9?) 2 9
Round the quotient 26/3 to two decimal places. 26 ÷ 3 = 8.67 To get the calculator out of rounding mode, use: MODE (Norm) 3 (Norm 1 ∼ 2?) 1
How do we estimate operations with decimals? To estimate the result of an operation with decimals, round the numbers to the nearest integers and then perform the operations. 5.23 · 6.8 ≃ 5 · 7 = 35, while 5.23 · 6.8 = 35.564 35.874 : 9.15 ≃ 36 : 9 = 4, while 35.874 : 9.15 = 3.92 26
UNIT 4
Method for solving word problems To solve problems involving decimal numbers, apply the following four steps: 1. Data 2. Question
3. Approach and working out
4. Solution and verification 10
A group of 24 students is going on a field trip. They buy a case with 24 soft drinks for €7.28 and 24 sandwiches for €25.60. How much does each student have to pay?? 1. Data • 24 students • 24 soft drinks for €7.28 • 24 sandwiches for €25.60 2. Question: • How much does each student have to pay?
3. Approach and working out: Each student pays = total expenses: number of students Total expenses: €7.28 + €25.60 = €32.88 Each student pays: €32.88 : 24 = €1.37
4. Solution: Each student pays €1.37. The solution is reasonable and is expressed in Euros. Verification: €1.37 · 24 = €32.88 Geometrical problem 11
The perimeter of an equilateral triangle measures 8.54 m. Find out the length of each side, rounding the result to two decimal places. 1. Data • Perimeter 8.54 m 2. Question: • Find out the length of each side. Round the result to two decimal places. 4. Solution: Each side measures 2.85 m.
18. Round the numbers to two decimal places mentally. a) 23.7688
b) 4.4528
c) 5.8746
19. Estimate the result of the following operations.
Then find the exact value with a calculator to check the result. a) 13.95 + 22.05 b) 18.78 – 5.85 20. Estimate the result of the following operations. Then find the exact value with a calculator to check the result. a) 8.92 · 7.12 b) 24.88 · 4.93 c) 56.87 · 10.15
3. Approach and working out: Length of one side = perimeter divided by 3 8, 5 4 3 25 2,8 4 6 1 4 2 0 Rounding: 2.85 m 2
21. Solve the problems and round the answers to two
decimal places. a) 688.567 + 567.4 b) 45.894 – 9.823 c) 6.65 · 5.4 d) 34.56 : 4.2
22. Two people form a company. Each person owns
half of it. The first year, they make 37 000 € in profit. How much of the profit does each person receive? Give your approximated answer in whole euros.
Decimal numbers
27
UNIT
5
Powers and square roots 1 How do we work with powers? What are powers?
Special cases
03= 0 03 = 0 · 0 · 0 = 0 0n = 0, n ≠ 0 14 = 1 14 = 1 · 1 · 1 · 1 = 1 1n = 1
23 = 2 · 2 · 2 = 8 2 is the base and 3 is the exponent
A power is a product of equal factors. Exponent
n
an = a · a · … · a
Base
The base of a power is the factor that you multiply. The exponent is the number of times that you multiply the base by itself. If the exponent is 2, we say the number is squared. If the exponent is 3, we say the number is cubed. If the exponent is 4, we say ‘to the fourth� or ‘to the power of four�; if 5, we say ‘to the fifth� or ‘to the power of five�; and so on. Scientific notation
Light-year = = 9.461 · 1012 km = = 9 461 000 000 000 km Radius of an oxygen atom = = 6.6 · 10 –11 m = = 0.000000000066 m
The scientific notation of a number is the expression of it as the product of a decimal number in which the integer part is a single nonzero figure with an integer power of 10 Scientific notation is used to represent very large or very small numbers. Products of powers with the same base The product of two powers with the same base is another power with the same base, but the new exponent is the sum of the two exponents. an · ap = an + p
Special cases
• Any number other than zero to the zero power is equal to one. 50 = 1 Verification: 3 50 = 53 = 125 = 1 5 125 • A number raised to the first power is equal to that same number. 51 = 5 Verification: 2 51 = 5 = 25 = 5 5 5 26
UNIT 5
3
53 · 54 = 57
4
Verification: 53 · 54 = 5 · 5 · 5 · 5 · 5 · 5 · 5 = 57
Quotients of powers with the same base The quotient of two powers with the same base is another power with the same base, but the new exponent is the difference between the two exponents. an : ap = an – p 57 : 53 = 54
7
Verification: 57 : 53 = 5 = 5 · 5 · 5 · 5 · 5 · 5 · 5 = 5 · 5 · 5 · 5 = 54 5·5·5 53
Powers of powers The power of a power is another power with the same base and an exponent that is the product of the exponents. (an)p = an · p
Rules of exponents Rules
3 2 2 2 (52)3 = 56 Verification: (52)3 = (52) · (52) · (52) = (5 · 5) · (5 · 5) · (5 · 5) = = 5 · 5 · 5 · 5 · 5 · 5 = 56
Examples
an · ap = an + p
57 · 54 = 511
an : ap = an – p
57 : 54 = 53
(an)p = an · p
(52)3 = 56
(a · b)n = an · bn (5 · 7)3 = 53 · 73
Powers of products
(a : b)n = an : bn (5 : 7)3 = 53 : 73
The power of a product is equal to the product of the factors raised to the same exponent. (a · b)n = an · bn 3 3 3 (5 · 7)3 = 53 · 73 Verification: (5 · 7)3 = (5 · 7) · (5 · 7) · (5 · 7) = (5 · 5 · 5) · (7 · 7 · 7) = = 53 · 73
0n = 0, n ≠ 0
03 = 0
1n = 1
13 = 1
a 0 = 1, a ≠ 0
50 = 1
a1 = a
51 = 5
Powers of quotients The power of a quotient is equal to the quotient of each of the numbers raised to the same exponent. (a : b)n = an : bn (5 : 7)3 = 53 : 73
()
3
3
3 3 Verification: (5 : 7)3 = 5 = 5 · 5 · 5 = 5 · 5 · 5 = 5 = 53 : 73 7 7 7 7 7 · 7 · 7 73
3
1. Write the numbers using scientific notation. a) 230 000
b) 0.00057
2. Write the numbers in scientific notation as decimal
numbers.
a) 5.6 · 103 b) 91
c) (– 6)1
d) (– 8)0
4. Write the result in the form of a single power using
the rules of exponents.
a) 35 · 34
b) 7 8 : 75
c) (34)2
d) 65 · 64 · 62
quotient in each of the following operations.
a) (2 · 5)3 b) (7 : 3)4 c) (3 · 7 · 13)5 d) (2 : 11)7 6. Apply the rules for the power of a product or a
quotient to obtain a result with only one power.
b) 7.95 · 10 – 3
3. Calculate mentally. a) 7 0
5. Apply the rules for the power of a product or of a
a) 83 · 73
b) 54 : 34
c) 35 · 25 · 55
d) 116 : 136
7. Apply the rules of exponents to obtain a result with only one power. a) x 3 · x 4
b) x 6 : x 2
c) (x 2)3
d) x 2 · x 3 · x 5
Powers and square roots
27
2 How do we work with square roots? What are square roots? √25 = ± 5
The square root of a number a is another number b, so that b squared equals a.
⎧52 = 25 because ⎨ 2 ⎩(– 5) = 25
√ a = b if b2 = a
√
Radical symbol
a
Radicand
b
Root
The geometric interpretation of the square root of a number consists of finding the length of one side of a square whose area is that number. 1
A gardener wants to plant a square flowerbed with an area of 25 m2 How long must each side be? √25 = 5, each side has to measure 5 m
25 m2
Number of square roots of a number A number can have two square roots, one or none. Convention
In square root operations, the root takes the sign (that is positive or negative) of the radical. √ 49 – √ 25 = 7 – 5 = 2
Radicand
Number of roots
Example
Check
0
One
√0 = 0
02 = 0
+
Two opposite
√9 = ± 3
32 = 9; (– 3)2 = 9
–
None
√– 9 is not a real number
Types of square roots A square root can be: √49 = ±7
• Perfect square: A square is perfect and called a ‘perfect square� or a ‘square number� when it is the product of an integer multiplied by itself, for example, 1, 4, 9, 16 • Imperfect square: A square is imperfect when the radicand is not a perfect square. In these cases, you can find the two integers that the square root is between. The smallest is called root by default, and the largest, root by excess. 2
Find the two natural numbers that the positive square root of 73 is in between. If you have 64 < 73 < 81, and then take the square roots of 64 and 81, you get 8 < √73 < 9 The closest lower square root is 8, since 82 = 64, which is less than 73 The closest higher square root is 9, since 92 = 81, which is greater than 73
28
UNIT 5
Finding square roots using a calculator Calculators have a square root button √– to find the square root of a number. This is how you use it. a) Newer calculators
Press the square root key, enter the radicand and press =
–
√289 √
b) Older calculators
Enter the radicand and press the square root key.
√289
289 = 17 –
289 √
17
What is the order of operations? When an expression with square roots and exponents has multiple operations, they must be performed in this order: a) Parentheses.
( ) n
an √
b) Exponents and square roots.
· :
c) Multiplication and division (from left to right).
+ –
d) Addition and subtraction (from left to right).
(75 – 82) · √36 = (75 – 64) · 6 = 11 · 6 = 66 Avoiding common errors a) Remember that √a + b is not equal to √a + √b b) Remember that √a – b is not equal to √a – √b
√9 + 16 = √25 = 5
√25 – 9 = √16 = 4 while
while √9 + =
√25 – √9 =
=3+4=7
=5–3=2
8. Calculate the square root of the perfect squares
mentally.
a) 25
b) 49
c) 0
d) 1
9. Calculate the default whole square root of: a) 53
b) 23
c) 17
d) 90
10. Calculate the whole square root by excess of: a) 45
b) 87
c) 15
d) 60
11. Use a calculator to find the square roots. a) 361
b) 441
c) 7 921
d) 710 649
12. Perform the calculations. a) (26 + 72 – 82) · √81
b) √49 + √64 : √16
13. In your notebook, complete the calculations with
= or ≠
a) √36 + 64
■ √36 + √64
b) √36 + 64
■ √100
c) √100 – 36 ■ √9 – √16 14. Create a problem demonstrating the square root
of 64 geometrically.
Powers and square roots
29
EXERCISE 1
Hosting an exchange student Lucía lives in Málaga and her family is going to host Olivia, a student from the USA who lives in New York. They are going to meet in Málaga in July, so that Olivia can practise her Spanish and visit some sites in Andalusia. Olivia has read that Málaga is a very touristic area. Lucía’s mother, María, told her that before the pandemic, Málaga received about 13 million tourists, with an economic impact of about 14 400 millones de euros. Lucía says that’s about 14.4 millardos and Olivia says it’s about 14.4 billones.
1 To understand what Olivia and Lucía say, look up the difference between the short and long number scales, and in which countries each scale is generally used. 2 Define «millardo»: Is it a numeral adjective or a noun? 3 Is it right to read the number 14 400 000 000 as 14 billion four hundred million?
If the economic impact is 1.44 · 109 and 1.30 · 107 tourists have visited the region, what is the average economic impact per person during their stay?
EXERCISE 2
4
Over lunch, Olivia says that the avocados in the salad are very tasty and asks if they are locally grown. Lucía’s father, Pedro, replies that the Axarquia region is the largest avocado producer at the national and European level, but that right now there is a severe water shortage and they will have to come up with strategies in order to maintain the avocado production. They have researched the following data: • A mapping of the crops in the Axarquia region has found that there is an irrigated area of 12 989.96 hectares with an average annual water consumption of 5 866 m³ per hectare. • La Viñuela reservoir, with a capacity of 165.43 hm³, is the main source of water in the region.
1
How many cubic hectometres will be used for irrigation over the course of a year?
2 It is estimated that 14.60 hm³ of water is destined for human consumption, 0.21 hm³ is destined for livestock consumption, and 0.82 hm³ is used to maintain golf courses. What is the total annual water consumption in the region? 3 The region has 224 000 inhabitants but the population increases during the summer months of June, July, August, and September in municipalities such as Nerja, Torrox, Torre del Mar and Rincón de la Victoria. According to the National Statistics Institute, the average water consumption per person over that period is 7.60 m³ per month. Taking this into account, estimate the population increase in the region. 4 If the reservoir is at one-fifth of its total capacity because of the drought, will there be enough water to maintain
current consumption levels?
32
EXERCISE 3 EXERCISE 4
Olivia enjoys playing golf. Her host family is planning to visit Benalmádena and play a few holes at a golf course. They have done some research and have decided to visit a course that charges €24 for adult visitors and €14 for children under 14. They want to play a Pitch & Putt, which is an 18-hole course where the distance between each hole is 90 m and the par for the course is 54 strokes.
1
How much will the two adults and the two girls, who are under 14, have to pay?
2
What is the par for one hole? Fill in the table and rank each player´s score. Player
Strokes
Score
Olivia
50
✎
Lucía
✎
–3
María
54
0
Pedro
✎
+4
One of the trips they want to take with Olivia is a day trip to the Caminito del Rey. This is a 7.7 km walk along paths and footbridges suspended at a height of 100 m. The visit takes them to the Desfiladero de los Gaitanes Natural Park, a limestone gorge created by the Guadalhorce River over millions of years.
The journey. The trip can be done by car, taking 54 minutes for the 59.3 km journey. An average car consumes approximately 7 L of petrol per 100 km. The price of petrol is €1.8/L. How much will be spent on fuel? 1
If the vehicle they are using emits 192 g of CO2 per passenger kilometre, what is the car’s carbon footprint for this trip, expressed in kilograms of CO2? They could also get there by train. The ticket price is €8 per person and the journey to the El Chorro station is 38.2 km. On arrival, they must take a bus to Ardales, where the Caminito starts. The bus from El Chorro to Ardales takes approximately 20 minutes to travel the 10 km distance. Then they must walk 2 km along a path until they reach the main gate. Bus tickets can be bought from the driver at a price of €1.55 per journey per person. Work out how much they would spend on fuel when travelling by car and how much they would spend if travelling by train. 2 Ticket price. The official website states that a general admission ticket costs €10. A ticket including an official guided tour costs €18. Work out how much they would pay if buying general admission tickets and how much it would be if they pay for the guided tour.
The other cost of the trip. CO2 emissions. The train emits 0.04 kg of CO2 per passenger kilometre. The bus emits 0.03 kg CO2 per passenger kilometre. Work out the carbon footprint of the train and bus journey and compare it to the carbon footprint of the car journey. Remember that some journeys are round trips.
EXERCISE 5
3
1
Organise a trip with activities you would like to do with your family or friends.
2 Prepare a budget. 3 Calculate your carbon footprint and assess the best way to reduce it.
Learning experience
33
What is an irreducible fraction and how is it obtained? Give an example. Solution A fraction is irreducible if it cannot be simplified. That is to say, the numerator and denominator are prime to each other. To calculate an irreducible fraction, the procedure is as follows: a) Find the GCD (greatest common divisor) for the numerator and denominator. b) Divide the numerator and denominator by the GCD. Whenever possible, simplify the fraction and make it irreducible. Example: 12 = 12 : 6 = 2 18 18 : 6 3 1
GCD(12,18) = 6
2
Work out the factorial decomposition of the following numbers: a) 540 b) 1 800
Work out: a) 5 – 2 · (3 + 2) – 4 · (4 – 7) Solution a) 7 b) – 13
Solution a) 540 = 22 · 33 · 5 b) 1 800 = 23 · 32 · 52
3
4
b) – 7 + 12 : (17 – 14) – 5 · 2
Work out: a)
4 · 1 – 4 5 6 3
b)
5 · (2 – 5 ) + 7 4 3 6
5
Work out: a) 34.876 + 803.25 + 650.94 b) b) 46.431 : (25.63 – 18.7)
6
Solve the following operations:
a) _24 + 52 – - 72 i $
64
b)
19 12
Solution a) 34.876 + 803.25 + 650.94 = 1.489,066 b) 46.431 : (25.63 – 18.7) = 46.431 : 6.93 = 6.71
b) _ 49 +
25 i| 16
Solution a) _24 + 52 –- 72i $ 64 = – 8 · 8 = – 64 b) _ 49 + 25 i| 16 = (7 + 5) : 4 = 12 : 4 = 3 34
Solution a) – 6 5
To pay for a trip, we divide the total cost into three monthly payments of €749 plus a €6 administration fee. How much do we pay in total for the trip? Solution 749 · 3 + 6 = €2 253 7
A square plot of land has an area of 756.25 m2. How much does it cost to fence the plot if each metre costs €5? Solution Length of one side: L = √756.25 = 27.5 m Perimeter = 4 · 27.5 = 110 m Cost = 110 · 5 = € 550 8
1. What is rounding and how is it carried out? Give an example. 2. Work out the GCD (greatest common divisor) and the LCM (least common multiple)
of 780 and 600
3. Work out: a) 5 – (– 3 + 4 – 2) – 3 · (2 + 5 – 4) b) – 23 + 7 · 8 – 3 · (5 + 8 – 18) 4. Work out:
(
a) 5 –
)(
3 : 19 + 7 4 12 6
)
b)
3 · 5 – 2 : 4 4 6 3 5
5. Work out: a) 725,36 – 584,258
b) (87,25 – 23,508) · 7,5
6. Solve the following operations: a)
25 + 81 $ 9
b)
81 : (62 – 33)
7. Oscar and Pablo make stopovers at the Paris airport. Oscar stops every 20 days and
Pablo every 24 days. They coincide on a certain day. How many days will it take them to coincide again? 8. To pack socks, we put each pair in a small cube-shaped box. Then, we put the small
boxes into larger boxes, so that 36 boxes of socks fit in the bottom of a large box and 6 boxes fit in each column. Write the total number of boxes of socks, expressed as a power. If each box of socks costs €5, what is the price of the large box containing the smaller boxes with the pairs of socks?
Evaluation
35
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