Mathematics 1 Key concepts is a collective work, conceived, designed and created by the Secondary Education department at Santillana, under the supervision of Teresa Grence. WRITERS José Antonio Almodóvar Ana María Gaztelu Augusto González Pedro Machín Silvia Marín Carlos Pérez Domingo Sánchez EDITORS José Antonio Almodóvar Isabel Checa Silvia Marín CLIL CONSULTANT Guillermo Dierssen EXECUTIVE EDITORS Nuria Corredera Carlos Pérez PROJECT DIRECTOR Domingo Sánchez BILINGUAL PROJECT DIRECTOR Margarita España Do not write in this book. Do all the activities in your notebook. SECONDARY 1 Mathematics Key concepts
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Mathematics 1 Key concepts is a collective work, conceived, designed and created by the Secondary Education department at Santillana, under the supervision of Teresa Grence.
WRITERS José Antonio Almodóvar Ana María Gaztelu Augusto González Pedro Machín Silvia Marín Carlos Pérez Domingo Sánchez
EDITORS José Antonio Almodóvar Isabel Checa Silvia Marín
CLIL CONSULTANT Guillermo Dierssen
EXECUTIVE EDITORS Nuria Corredera Carlos Pérez
PROJECT DIRECTOR Domingo Sánchez
BILINGUAL PROJECT DIRECTOR Margarita España
Do not write in this book. Do all the activities in your notebook.
The base-ten system uses ten different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is a positional system, meaning the digits have different values depending on their position within the number.
Example: 13 460 090 = 1 T. of millions + 3 U. of millions + + 4 H. of thousands+ 6 T. of thousands + 9 Tens
The Roman numbering system uses seven different letters:
I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1 000
EXAMPLE
1. Write down the value of each number.
a) MDXXIII = 1 523 b) CDXLI = 441 c) CXCIX = 199
Truncating a number to a certain order consists of replacing the digits of the lower orders with zeros.
In order to round a number to a certain order, we look at the digit of the following order:
• If it is higher than or equal to 5, we add 1 to the digit we are rounding to.
• If it is lower than 5, we leave the digit as it is.
Finally, we truncate the number we have obtained.
EXAMPLE
2. Round the number 5 178 463 to the nearest thousand.
5 178 463 4 < 5" Rounding = 5 178 000
ACTIVITIES
1 Rewrite the following in the base-ten system.
a) XXII b) DCLXIII
2 Truncate and round these numbers to the nearest hundred.
a) 3 729 b) 653 497
1 Natural numbers
The digit 0 is used to indicate a number has no units of the order where the 0 is placed.
Multiplication rule for Roman numerals.
If a Roman numeral has a line over it, its value is multiplied by a thousand.
3 Calculate the dividend of a division where the divisor is 14, the quotient is 23 and the remainder is 2.
4 Compute and indicate how the following are read.
a) 24 b) 33
The distributive property also applies to subtraction.
3 ? (12 - 4) = 3 ? 12 - 3 ? 4
Properties of operations with natural numbers
3
3.1. Properties of the sum and multiplication
• Commutative property. The order of the addends or the products do not vary the result.
• Associative property. The order in which the summations are performed does not affect the result. The same thing applies for multiplication.
• Distributive property of multiplication over addition. A number multiplied by a sum of numbers is equal to the sum of the number multiplied by each of the addends.
3.2. Properties of subtraction and division
In a subtraction, the subtrahend plus the difference is equal to the minuend.
In a division, the dividend D is equal to the divisor d multiplied by the quotient c plus the remainder r, and the remainder must be smaller than the divisor.
D = d ? c + r r < d
Powers of natural numbers4
A power is an abbreviated way of writing a multiplication of equal factors.
an = a ? a ? a ? a ? … ? a 1444442444443
n times
a " is called the base and is the repeated factor.n " is called the exponent and indicates the number of times that the base is repeated.
The powers with an exponent equal to 2 are referred to as 'squared', and those with an exponent equal to 3 are referred to as 'cubed'.
EXAMPLE
3. Write down each power.
a) Seven to the power of six. 76 b) Five to the power of seven. 57
Powers of base 10. Polynomial decomposition of a number
5
A power of base 10 that has a natural number as its exponent is equal to the unit followed by n zeros, where n is its exponent. Example: 105 = 10 ? 10 ? 10 ? 10 ? 10 = 100 000 144444424444443 1442443 5 times 5 zeros
The polynomial decomposition of a number is equal to the sum of the products of its digits multiplied by a power of base 10 corresponding to its order. Example:
The exact square root of a number a is a number b, so when b is squared we obtain a.
a = b, when b2 = a
The radicand is the number a,
is the symbol of the root and we say b is the square root of a.
The numbers with an exact square root are called perfect squares.
a b=Symbol of the root
Radicand
RootF F
F
EXAMPLE
7. Calculate the roots of these perfect squares.
a) 4 = 2, as 22 = 4. b) 36 = 6, as 62 = 36.
7.2. Integer square root
If the radicand is not a perfect square, the square root is an integer square root.
The integer square root of a number a is the largest number b whose square is less than a. The remainder of the integer square root is the difference between the radicand a and the integer square root b squared.
Remainder = a - b2
ACTIVITIES
8 Compute these exact square roots.
a) 121 b) 144
9 Find the value of a in these non-exact square roots.
a) a . 5 and the remainder is 7.
b) a . 7 and the remainder is 3.
Calculate the square root of a number
Calculate the square root of 39.
Follow these steps
1. Find the largest number that when squared is less than, or equal to, the radicand.
52 = 25 " 25 < 39
62 = 36 " 36 < 39
72 = 49 " 49 > 39
2. If the square of the number is less than the radicand, that number is the integer square root. The difference between the radicand and the square of that number is the remainder.
b) 62 = 36 " 36 < 39
6 is the largest number whose square is less than 39.
The integer square root is 6 and the remainder is: 39 - 62 = 39 - 36 = 3
KNOW HOW TO
Taking the square root of a number and squaring it are inverse operations to each other.
13 Write down the following in the base-ten system.
a) XVIII b) LXXI c) XCVII d) MDCXXVIII
14 Approximate these numbers by truncating and rounding them to the nearest hundred.
a) 24 536 b) 200 664 c) 456 283
15 Apply the distributive property and compute.
a) 2 ? (5 - 3) b) (12 - 7 + 3) ? 8
16 Write down how the following powers are read.
a) 32 b) 75 c) 43 d) 1417
17 Write down the numbers whose polynomial decomposition is given by:
a) 6 · 104 + 7 · 103 + 9 ? 10 + 7
b) 3 · 105 + 4 · 102 + 1
c) 8 · 103 + 102
d) 2 · 106
18 Write down the following as a single power.
a) 24 ? 26 : 27 c) 102 ? 106 : 103
b) 35 : 33 ? 32 d) 76 : 73 ? 74
19 Write down the following as a single power.
a) 52 ? 32 c) 86 : 26 e) 210 ? 1010
b) 47 ? 27 d) 207 : 107 f ) 124 : 44
20 Compute.
a) (24)3 b) (52)5 c) (34)6 d) (75)3
21 Compute the integer square root and remainder for the numbers written down by Anne.
22 Perform the following operations.
a) 28 - 3 ? 2 ? 4 d) 14 : 2 + 3 ? 9 - 5
b) 5 ? 9 : 3 + 7 e) (42 - 6) : 6 + 5 ? 3
c) 25 + 4 ? 2 - 7 ? 3 f ) 15 ? (7 - 3) : (3 - 1)
23 Compute.
a) ?3 9 3 33 2 3- -
b) :?12 3 25 3 492+ +_ i
c) :7 64 5 52 3+ -
a) 79
b) 32
c) 140
d) 853
24 A boat was carrying 502 people. It made three stops. On the first stop 256 people left the boat. On the second stop 162 people got on the boat and on the third stop 84 people got off the boat. How many people are left on the boat after the three stops?
25 In one piggy bank there are 246 € and in a different one there are 114 €.
a) If all the money is in 2 € coins, what is the total number of coins in the piggy banks?
b) If all the money was in 5 € notes, how many would there be?
26 How much money is there in a wallet containing two 20 € notes, three 10 € notes, six 5 € notes and four 2 € coins?
27 Six people have 1 000 € to spend on a trip. They must travel by both plane and train. The train ticket costs 38 € and the plane ticket costs 125 €. Do they have enough money to go on the trip?
28 In an ethnic music festival there are artists from three different continents. 350 come from Asia. There are 157 more artists from Africa than there are from Asia and 98 fewer artists from Europe than there are from Asia. What is the total number of artists?
29 An orange tree has produced 40 kg of oranges this year but only 27 kg last year. If last year the price per kilogram was 3 € and this year it is 2 €, have profits increased or decreased compared to last year?
30 Raquel had 12 €. She spent half at the cinema and the other half on a winning lottery ticket that won her 15 € for each euro she paid for it. How much money did she win?
31 Two flowers cost 3 € and to make a bouquet you need 12 flowers.
a) How many bouquets can you make for 90 €?
b) If you want to make a profit of 40 €, how much should you charge for each bouquet?
9 Is there a divisibility relation between the following pairs of numbers?
a) 135 and 45 b) 238 and 16
10 Which of these series is formed by multiples of 4? And by multiples of 5?
a) 1, 4, 9, 16, 25… d) 4, 8, 16, 24, 32, 40…
b) 5, 10, 15, 20… e) 1, 5, 10, 20, 30…
c) 8, 10, 12, 14, 16… f ) 20, 40, 60, 80…
11 Find three numbers that are multiples of both 6 and 5. Are they multiples of 10 too?
12 Given the relation 104 = 4 ? 26, which of the following statements are true?
a) 104 is divisible by 4. c) 26 is a divisor of 104.
b) 104 is a multiple of 4. d) 104 is divisible by 26.
13 The number a is divisible by 4. Calculate a if the quotient of the division is 29.
14 Are all numbers divisible by 2 also divisible by 4? And vice versa?
15 Copy and complete the table in your notebook.
Number Divisors Prime/Composite
17
29
58
72
97
113
16 If the division a : 4 is exact, is a a prime or a composite number?
17 Decompose the following numbers into their prime factors.
a) 560 b) 2 700 c) 616 d) 784 e) 378 f ) 405
18 Ben and Julie have factorised the number 2 250, obtaining the following results.
Ben: 2 ? 32 ? 53 Julie: 32 ? 52 ? 10
Are their results correct?
19 Group up factors and write down the following factorial decompositions correctly.
a) 22 ? 3 ? 2 ? 33 c) 32 ? 5 ? 5 ? 32
b) 52 ? 7 ? 54 ? 73 ? 7 d) 22 ? 7 ? 2 ? 72 ? 2
20 Calculate the greatest common divisor and the lowest common multiple of the following groups of numbers.
a) 10, 20 and 100 b) 5, 9 and 45 c) 4, 30 and 50
21 There are 18 000 plates in a warehouse. The company decides to put them in boxes. Each box contains a dozen plates.
a) How many boxes will they need?
b) If the number of plates in the warehouse was tripled, how many boxes they need?
c) If they could only fit half a dozen plates in each box, how many boxes would they need?
22 A stationery shop sells pencils in boxes of 8, 10 or 15. How many boxes of each size can it sell if there are 270 pencils and all of the boxes are of the same size? Will it sell all the pencils in each case?
23 Tania had a bag containing 35 sweets. She ate some of them, but does not remember how many. However, she knows she can group the ones she has left into bags of 2, 3 and 5 sweets without any remaining.
a) How many sweets does she have left?
b) If she groups them into bags of 2, how many bags will she need?
c) What about if she groups them into bags of 3? And into bags of 5?
24 Alice wants to put 45 books in a bookcase. She would like each shelf to contain the same number of books.
a) How many books can she put on each shelf?
b) How many shelves will she need in each case?
25 Helen and Martha have a collection of stickers. Helen counts them 7 by 7 and Martha 4 by 4. What is the minimum number of stickers their collection can have?
26 Rachel and David both go horse riding. Rachel goes every 3 days and David every 4 days. If they meet on the 24th February, when will they next meet? How many times will each of them have gone horse riding before?
1 Is there a divisibility relation between the following pairs of numbers?
a) 4 and 18 c) 3 and 4
b) 5 and 30 d) 7 and 91
2 Which of the following numbers are divisible by both 8 and 12?
a) 288 c) 576 e) 480
b) 364 d) 1 248 f ) 672
3 Calculate the divisors of these numbers.
a) 75 c) 81 e) 121
b) 77 d) 96 f ) 113
4 Lucas has made 45 cakes and wants to store them in boxes. He wants each box to contain the same number of cakes. In how many different ways can he store them so there are none remaining?
Prime and composite numbers
5 Which of the following numbers are prime and which are composite? Write down three divisors of the composite numbers.
a) 133 c) 179 e) 210
b) 153 d) 184 f ) 301
Decomposition into factors
6 Decompose the following numbers into their factors.
a) 240 b) 345 c) 99 d) 5 700
7 To which numbers do these prime decompositions correspond?
a) 2 ? 32 ? 5 b) 24 ? 5 c) 3 ? 52 d) 2 ? 32 ? 7
Greatest common divisor and lowest common multiple
8 Calculate the greatest common divisor of these numbers.
a) 45 and 75 c) 24, 66 and 84
b) 16 and 24 d) 72, 108 and 144
9 Calculate the lowest common multiple of these numbers.
a) 18 and 24 c) 12, 18 and 60
b) 15 and 25 d) 15, 25 and 45
10 Carol has 40 European stamps and 56 Asian stamps. She wants to make the smallest possible number of equal sets of stamps. She does not want to mix any European stamps with Asian stamps or have any left over. How many sets will she make? How many stamps will each set have?
The absolute value of an integer is the distance, in units, that separates it from the number zero in the number line. It is written as ; ;, and is equal to the number without its sign.
;+b; = b ;-a; = a
The absolute value of 0 is 0, ;0; = 0.
1.2. Opposite of an integer
We say two different integers are opposites of each other if they are the same distance from 0.
Op (+a) = -a Op (-a) = +a
3 Integers
The number 0 is the only integer that is neither a positive nor a negative integer.
ACTIVITIES
1 Calculate the absolute value of the following numbers.
a) ;+3;
b) ;-2;
c) ;-5;
2 What are the opposite integers of these numbers?
a) Op (+2)
b) Op (-6)
c) Op (-8)
3 Write down the following integers from largest to smallest.
-6, +7, -10, -4, +8
Ordering integers2
• Any positive integer is larger than any negative integer.
• When there are two positive integers, the largest is the one with a larger absolute value.
• When there are two negative integers, the largest is the one with a smaller absolute value.
2. Give the result a positive sign when multiplying two numbers that have the same sign, and a negative sign when the two numbers have different signs.
EXAMPLE
4. Compute these multiplications.
a) (-3) ? (+5) = -15 b) (-7) ? (-2) = +14
F
Different sign F
Same sign
5.2. Division of integers
To divide two integers:
1. Divide their absolute values.
2. Give the result a positive sign if they have the same sign, and a negative sign if their signs are different.
EXAMPLE
5. Compute these divisions.
a) (-12) : (-4) = +3 b) (-9) : (+9) = -1
F
Same sign F
Different sign
Multiply and divide multiple integers
Compute the value of the following expression: (-15) : (+5) ? (+2).
Follow these steps
1. Calculate the sign of the result of the operation. (-15) : (+5) ? (+2) = -
2. Multiply or divide, from left to right, the absolute values of the numbers. Then add the sign calculated above.
When dealing with addition, subtraction, multiplication and division of integers in one expression, we must perform these operations in the following order:
1. Start by computing multiplications and divisions from left to right.
2. Compute the remaining additions and subtractions from left to right.
EXAMPLES
6. Perform the following operation.
(-10) : (+2) - (-4) ? (+1) =F F Multiplications and divisions
17 Write down the integer corresponding to each of the points marked in the number line.
a) A B C D
0 1
b) A B C D
0 1
18 Order the following numbers from smallest to largest.
4 -6 -7 2 -9 -11 12 -8 16 -19
Order their opposites and absolute values too.
19 Eliminate the brackets and compute the value of the following.
a) -(-8) + 17 e) -(-9) + 15
b) -(-5) - 23 f ) -(-12) - (-20)
c) -(-30) + (-12) g) -(-24) + (-19)
d) -(-6) - (-18) h) -(-16) - (-14)
20 Compute the following additions and subtractions of integers.
a) (-3 + 8) - (-2) e) (5 - 9) + (-3)
b) (-2 + 4) - (+4) f ) (7 - 10) + (-8)
c) (1 - 3) + (-4) g) (-2 - 8) - (+6)
d) (6 - 3) + (-2) h) (-1 + 4) - (-7)
21 Perform the following operations.
a) –1 – [–3 – 2 + (–4)]
b) (2 – 8) + [1 – (–9) – 3]
c) (–5 + 3) – [7 – (–8)]
d) –6 – [5 – 10 + (–3)]
e) –4 + [–6 + (–2) – (–5)]
f ) 12 + (–9) – [(–7) – (+5)]
g) –[5 + (–18) + (–6 – 12) – 3]
h) 4 + [–6 + (–9) – 12]
22 Calculate.
a) (+12) : (+3) d) (+6) ? (-8)
b) (+15) : (-3) e) (-12) ? (-3)
c) (-28) : (-7) f) (-7) ? (+10)
23 Calculate.
a) (7 - 10) ? (1 - 6) e) (-3 + 9) ? (4 - 2)
b) (5 - 12) ? (-3 + 5) f ) (-1 - 3) ? (9 - 7)
c) (-15 + 3) : (-7 + 4) g) (9 - 18) : (6 - 3)
d) (-12 - 6) : (-1 - 2) h) (-8 + 16) : (-4 + 6)
24 An aeroplane flies at a height of 7 950 metres above sea level and a submarine is directly beneath it, at 275 m below sea level. How many metres are there between them?
25 A mountain climber reaches the peak of a mountain at a height of 2 532 m. A miner is digging directly beneath him at a depth of 180 m.
a) Express these measurements as integers.
b) How many metres are there between them? Compute this using an operation between integer numbers.
26 The temperature in a room increases by 30 °C and then decreases by 42 °C. If the final temperature is -6 °C, what was the initial temperature?
27 Peter and Louise have a savings account where they receive their salaries and pay their expenses. These are the latest movements in their account.
Movement Balance Description
-120 200 Electricity bill
1 500 Peter’s salary
1 400 Gas bill
-1 470 Mortgage
730 Louise’s salary
a) What was their balance before paying the electricity bill?
b) What is their balance after receiving Peter’s salary?
c) How much was the gas bill?
d) What is their balance after paying their mortgage?
e) How much is Louise’s salary?
28 Alexander works on the 23rd floor of a building. When he parks his car in the company’s carpark, he must go up 27 floors to get to his office. On what floor does he park his car?
29 Euclid, a famous mathematician, died in the year 265 BC and lived for 60 years.
a) What year was he born?
b) How much older than you is Euclid?
c) In which year was a person who is two years older than Euclid born? EUCLIDES
a) How many numbers are there between -50 and +128?
b) What about between -48 and 48?
c) What integer has 9 as its opposite?
d) What integers have the same absolute value?
2 Write down the sets corresponding to the following.
a) Numbers greater than -7 and smaller than -2.
b) Numbers greater than -4 and smaller than +2.
c) Integers which are at a distance of 7 units from 3.
d) Integers whose absolute value is less than 6.
3 Order the following integers from smallest to largest.
Op (+5) -8 Op (-3) ;-4; +6
Operations with integers
4 Perform the following operations.
a) 6 + (-4 + 2) - (-3 - 1)
b) 7 - (4 - 3) + (-1 - 2)
c) 3 + (2 - 3) - (1 - 5 - 7)
d) -8 + (1 + 4) + (-7 - 9)
e) 10 - (8 - 7) + (-9 - 3)
f ) 7 - (4 + 3) + (-1 + 2)
5 Compute the following.
a) 5 - (-3) + 7 + (-9) - 14
b) 12 : (-5 + 3) - 4 ? (4 - 9) ? (-1)
c) 28 - 3 ? [(4 - 6 + 7) + (-5) ? (-4)]
6 Calculate the following.
a) 8 - 6 ? 3 + 12 : 2 - 5
b) 8 - 6 ? 3 + 12 : (2 - 5)
c) 8 - 6 ? (3 + 12 : 2) - 5
d) 8 - 6 ? (3 + 12 : 2 - 5)
e) (8 - 6 ? 3 + 12) : 2 - 5
7 Complete the following in your notebook.
a) 13 ? (4 - 8) = -26
b) 15 - 4 ? 3 = 33
c) 7 + (9 - 4 - 10) ? (-2) = 1
8 In a biology lab, biologists are studying the resistance of a microorganism to changes in temperature. They have a sample at -3 °C. They raise its temperature to 40 °C, then decrease it by 50 °C and finally raise it by 12 °C. What is the final temperature of the sample?
9 Xavier owed his brother 24 €. As he only had 15 €, he asked his friend Lucy for a loan. Lucy loaned him some money. With that money, Xavier paid back his brother and then bought 3 notebooks costing 2 € each. After that, he had 5 € remaining. Which of the following expressions can be used to find out how much money Lucy loaned Xavier?
Richmond is an imprintof Santillana Educación, S. L.
Printed in Spain
Richmond58 St Aldate'sOxford 0X1 1STUnited Kingdom
CP: 146700
All rights reserved. No part of this book may be reproduced, stored in retrieval systems or transmitted in any form, electronic, mechanical, photocopying or otherwise without the prior permission in writing of the copyright holders. Any infraction of the rights mentioned would be considered a violation of the intellectual property. If you need to photocopy or scan any fragment of this work, contact CEDRO (Centro Español de Derechos Reprográficos, www.cedro.org).