METHOD CONCEPT / EDITOR
Boom voortgezet onderwijs
AUTHORS
Benjamin del Canho
Maartje Elsinga
Jacqueline Kooiman
Gijs Langenkamp
Florine Meijer
Chantal Neijenhuis
Willem Schaap
Renee Springer
KERN WISKUNDE
HAVO / VWO
YEAR 1-A
BOOM VOORTGEZET ONDERWIJS
©
2022 Boom voortgezet onderwijs, Groningen, The Netherlands
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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, recording or otherwise without prior written permission of the publisher.
isbn 978 94 9322 495 7 www.boomvoortgezetonderwijs.nl
KERN Mathematics is an RTTI-certified method, which consists of four types of questions:
r Reproduction questions
t1 Training-related application questions
t2 Transfer-related application questions
i Insight questions
See www.docentplus.nl for more information regarding the RTTI-certification.
Translation
Anne van Eeden Marieke Spijkstra Eline Wilhelm
Book design & technical drawings
René van der Vooren, Amsterdam
Layout & technical drawings PPMP, Wolvega
1 Numbers
mathematics in daily life Counting with zeros and ones 8
1.1 Positive and negative numbers 10
1.2 Multiplying and dividing negative numbers 16
1.3 Fractions 22
1.4 Multiplying and dividing fractions 28
1.5 Powers 34
Test preparation 40
2 Plane figures
mathematics in daily life From tessellation to art and tiling patterns 44
2.1 Points and lines 46
2.2 Angles 52
2.3 Plane figures 58
2.4 Drawing and reflecting 64
2.5 Symmetry 70
Test preparation 76
3 Relations
mathematics in daily life Where am I on Earth? 80
3.1 The coordinate system 82
3.2 Relationships 88
3.3 Formulas and graphs 94
3.4 Linear and quadratic relationships 100
3.5 Variables 106
Test preparation 112
4 Processing data
mathematics in daily life How to win an election 116
4.1 Data 118
4.2 Ratios 124
4.3 Percentages 130
4.4 Graphs and charts 136
4.5 Relationships and graphs 142
Test preparation 148
Index 150
Numbers
In this chapter you will learn about different types of numbers and how to perform calculations using these numbers. You will also learn about magic squares, number series, and number pyramids.
The chapter ends with the legend about the origin of chess.
MATHEMATICS IN DAILY LIFE
Counting with zeros and ones 8
1.1 Positive and negative numbers 10
1.2 Multiplying and dividing negative numbers 16
1.3 Fractions 22
1.4 Multiplying and dividing fractions 28
1.5 Powers 34
Test preparation 40
Mathematics in daily life
GOAL You will learn how computers can perform calculations using only zeros and ones.
Counting with zeros and ones
Decimal numerals Do you ever stop to think about how easy it is to count and perform calculations owing to the fact that you can write down all numbers using the digits 0 through 9?
Numbers written using these digits are called decimal numerals: decimal means ‘based on the number ten’. In the past, some cultures used different methods to write numbers. For example, Romans used Roman numerals and Egyptians used hieroglyphs.
In the decimal system, the position of a digit within a number determines the value of that digit. From right to left, the value of a digit is multiplied by ten for every step to the left. For example, the 4 in 2104 represents 4 units, the 0 represents 0 tens, the 1 represents 1 hundred and the 2 represents 2 thousands 4 + 0 + 100 + 2000 = 2104.
b inary numbers Computers do not use decimal numerals. Therefore, they store numbers in a different way. Computers can only use the digits 0 and 1. This is due to the fact that computers use electricity, which can only be turned on (1) or off (0). Of course computers would be quite useless if they could not somehow perform calculations with larger numbers. This is done by writing the zeros and ones in a specific order:
0 = 0 4 = 100
1 = 1 5 = 101
2 = 10 6 = 110 3 = 11 7 = 111
In this case, only two digits are available to write a number. Therefore, they are called binary numbers: binary means ‘composed of two things’. Making larger numbers using only zeros and ones works in a similar way to making larger decimal numerals. The position of a digit within a number determines its value. Again, the value of a digit increases from right to left. Instead of being multiplied by ten, the value is multiplied by two for every step to the left. Therefore, the 0 in 110 represents 0 units, the 1 (middle) represents 1 two, and the 1 (left) represents 1 four: 110 0 + 2 + 4 = 6.
b its and bytes Decimal numerals are shorter than binary numbers. For example, 2104 written as a binary number is 100000111000. Modern memory chips can store tens of billions of zeros and ones. A piece of memory used to store one 0 or 1 is called a bit. Bits are organised in groups of eight. These groups are called bytes. A byte that stores 00000101 represents number 5 in the decimal system. A byte can store any value from 0 (00000000) to 255 (11111111).
A memory chip that can store 4 billion bytes (4 gigabytes).
Questions about the text
1 a What are binary numbers? R
b What is a bit? R
c What is a byte? R
2 Why do computers use binary numbers instead of decimal numerals? R
3 Write the following binary numbers as decimal numerals: T1
a 1000 c 1011
b 1001 d 1010010
4 a Write the decimal numerals from 8 to 16 as binary numbers. T1
b Write the decimal numerals 32, 64 and 128 as binary numbers. T2
c Write the decimal numerals 31, 63 and 127 as binary numbers. T2
In-depth questions
5 a Write 101 as a decimal numeral and multiply the result by 2. Write the result as a binary number. What do you notice?
Hint: 2 is equal to 10. T2
b Write 101 as a decimal numeral and multiply the result by 8. Write the result as a binary number. What do you notice?
Hint: 8 is equal to 1000. T2
6 a Multiply the binary numbers 1101 and 1000 without using decimal numerals. I
b What goes wrong when you multiply the bytes 10101001 and 00000010 and you want to write the result as a byte? I
research assignment
7 Long rows of zeros and ones are often hard to read for people. Therefore, the computer industry often uses hexadecimal numbers. Use the internet to answer the following questions:
What digits do hexadecimal numbers consist of?
How can a hexadecimal number be converted to a decimal numeral and vice versa? Give a few examples.
How can a hexadecimal number be converted to a binary number and vice versa? Give a few examples.
Why does every byte consist of 2 hexadecimal digits?
Present your findings in one of the following ways: presentation, news article, video, poster or quiz. Use at least three different sources for this assignment. I
Have you reached your goal?
R I know how decimal and binary numbers are built up.
T1 I can convert a decimal numeral to a binary number and vice versa.
T2 I can perform calculations with bytes.
I I understand what happens when two bytes are added together.
bu LA r Y decimal numeral bit binary number byte
VOCA
Positive and negative numbers
GOAL You will learn what positive and negative numbers are and how to add and subtract them.
Positive and negative A number line with integers is shown on the right. Numbers greater than 0 are positive and numbers less than 0 are negative. The number 0 is neither positive nor negative. Positive integers are also called whole numbers −3 and 3 are opposite numbers, just like 10 and −10. Opposite numbers are the same distance from 0 on the number line:
–3 and 3 are opposite numbers
Decimals In numbers with a decimal point, the digits to the right of the decimal point are called decimals. These decimal numbers lie in between the integers on a number line. For example, 2.4 lies between 2 and 3. And −1.75 lies between −2 and −1.
Greater than and less than The further you move to the right on a number line, the greater the numbers. And the further you move to the left, the smaller the numbers. For example, 5 is greater than 3 and −7.8 is less than −7.6. This can be written as follows: 5 > 3 and −7.8 < −7.6.
examples
1 9 is less than 11, because 9 lies to the left of 11 on the number line. Therefore, 9 < 11.
2 −9 is greater than −11, because −9 lies to the right of −11 on the number line. Therefore, −9 > −11.
3 2.4 is greater than 2.2, because 2.4 lies to the right of 2.2 on the number line. Therefore, 2.4 > 2.2.
4 −2.4 is less than −2.2, because −2.4 lies to the left of −2.2 on the number line. Therefore, −2.4 < −2.2.
The Latin word ‘integer’ means whole.
EXERCISES — PRACTICE
Positive and negative
8 What is the temperature according to this thermometer? T1
9 a Take another look at the number lines shown at the top of the left page. Draw a number line from –4 to 4. Indicate 3 and the opposite of 3 in the same way as is shown in the example. T1
b How many integers does the number line show? R
c What integer is the opposite of 7? And what number is the opposite of −2.5? T1
d What number is the opposite of 0? Explain your answer. I
Greater than and less than
12 Choose the correct words. Use a number line if necessary. T1
a −4 is less / greater than 3, because −4 lies to the left / right of 3 on the number line.
b −5 is less / greater than −2, because −5 lies to the left / right of −2 on the number line.
c 51 is less / greater than 49, because 51 lies to the left / right of 49 on the number line.
d −13 is less / greater than −15, because −13 lies to the left / right of −15 on the number line.
13 Copy and fill in < or >. T1
a 2 … 5 e 4.72 … 4.7
b –3 … –2 f –2.7 … –2.9
c 0 … –7 g –99 … –101
d –9 … –11 h –101 … –102
14 a Between which two integers does −1.75 lie? T2
b Give two different numbers that are less than −2.57 and greater than −2.58. T2
c Write the following numbers in the correct order, from smallest to largest. T2
–2.2 / 4.5 / –1.1 / 3.49 / –3.51 / –1.099
Decimals
10 a What are decimals? R
b How many decimals does 13.05 have? T1
c How many decimals does –0.7843 have? T1
d Draw a number line from −5 to 5 and indicate the location of the following numbers: –4, –3, –1.5, 2, 3.5 and 4.75. T1
11 Draw a number line from −1 to 1 and indicate the location of the following numbers: –0.5, 0.8, 0.25 and –0.9. T2
POSITIVE AND NEGATIVE NUMBERS
Adding You can use the number line to help you add two numbers. Each number is represented by an arrow. See the number line on the bottom left. When you add two numbers you calculate the sum of these numbers. This is represented by the sum of the two arrows drawn above the number line. Draw the two arrows head to tail.
Adding a negative number yields the same result as subtracting the opposite number: 4 + −3 = 4 − 3.
examples
1 3 + –4 = 3 – 4 = –1
2 –5 + 2.5 = –2.5
subtracting When you want to calculate the difference between two numbers, you have to subtract two numbers. Calculating the difference works in the same way as addition, except you have to reverse the arrow that represents the second number.
Subtracting a negative number yields the same result as adding the opposite number: 4 − −3 = 4 + 3.
examples
1
EXERCISES
— PRACTICE
Adding
15 Calculate using a number line. T1
a 0.4 + 2 c –2 + 0.4
b –0.4 + 2 d –2 + – 0.4
16 Copy and fill in the blanks. T1
a 5 + –2 = 5 – 2 = …
b 8 + –10 = 8 … 10 = …
c 13 + –5 = 13 … 5 = …
d 2.5 + –7.5 = 2.5 … 7.5 = …
e 15 + –5.5 = …
f 1 + –6 = …
g 35 + –10 = …
h 11 + –20 = …
17 a Draw two number lines from −4 to 5 and use them to calculate −3 + 5 and 5 + −3.
What do you notice? T1
b Copy and fill in is or is not T2
When adding two numbers, the order of the numbers important.
18 Calculate. Use a number line if necessary or take another look at exercise 17. T1
a –2 + 25 e –9 + 7
b –3.5 + 7 f –18 + 9
c –7.5 + 16 g –19.5 + 11
d –19 + 31 h –13 + 7.5
19 Calculate. T1
a 10 + –5 e –8 + –17
b 4 + – 8 f –25 + –150
c –7 + 17 g 7 + –4.5
d –12 + 7 h –8.5 + –20
subtracting
20 Calculate using a number line. T1
a 5 – 3 c 3 – 5
b –5 – 3 d 3 – – 5
21 Use a number line to explain that order is important when subtracting two numbers:
2 − 7 = −5 and 7 − 2 = 5. I
22 Calculate. Use a number line if necessary. T1
a 13 – 17 e –1.5 – 4
b 7 – 22 f –9 – 12
c 9.5 – 12 g –25 – 16
d 5.5 – 14.5 h –8.5 – 2
23 Copy and fill in the blanks. T1
a 1 – –2 = 1 + 2 =
b 9 – –6 = 9 6 =
c –8 – –5 = –8 5 =
d –4.5 – –7 = –4.5 7 =
24 Calculate. T1
a 15 – 25 e 2.5 – 5
b –5 – –3 f –15.5 – 40
c –3 – 10 g 7.5 – –4
d 12 – –18 h –15 – –35.5
EXERCISES — EXPLORING & INVESTIGATING
exploring
25 Julius Caesar was born on 13 July of the year 100 BC and killed on 15 March of the year 44 BC. How old was Caesar when he died? T2
26 The average temperatures of the planets in our solar system are shown in the table below. T2
a On average, how many degrees warmer is Mars compared to Venus?
b On average, how many degrees colder is Neptune compared to Earth?
c On average, how many degrees warmer is Jupiter compared to Uranus?
Investigating
28 Read the text about Magic squares on the right page. Copy the following magic square and fill in the missing numbers. T2
9 −16
29 Design a magic square measuring 3 by 3. Use negative numbers as well as positive numbers. I
27 Copy and fill in the blanks. I
a 10 – = –2 e + 11.4 = 7
b 75 – = 125 f – –2.5 = –4.1
c + –69 = 73 g –1.35 + = 0.65
d – –12 = 6 h –0.99 – = 1.01
VOCA bu LA r Y
number line greater than integer less than positive adding negative sum whole number subtracting opposite numbers difference decimal magic square decimal number
Magic squares
Magic squares are squares with numbers, where the sums of the numbers in each row, column and diagonal are equal. Magic squares have been known for a long time and have triggered mathematicians’ and artists’ imaginations. A well-known magic square can be found on the wall of the famous Sagrada Família, a church in Barcelona designed by the Catalan architect Antoni Gaudí (1852−1926). The sum of the numbers in each row and column is 33. This number might refer to Jesus’ age at the time of his death, but it is was also considered important by the Freemasons, of which Gaudí was a member for quite a while.
The magic square on the Sagrada Família in Barcelona
COMP u TATIONAL s KILL s
C1 Mount Everest is the highest mountain in the world. Its peak has an altitude of 8848 metres above sea level.
I The peak of the second highest mountain in the world, K2, is 237 metres lower than Mount Everest’s peak. What is the altitude of K2’s peak?
II The peak of the third highest mountain in the world, Kangchenjunga, is 25 metres below K2’speak. Calculate the difference in altitude between the peaks of Kangchenjunga and Mount Everest.
Have you reached your goal?
R I know what positive and negative numbers are.
T1 I can place numbers on a number line and I can add and subtract positive and negative numbers.
T2 I can set up a calculation based on a word problem.
I I can make a magic square that includes negative numbers.
Multiplying and dividing negative numbers
GOAL You will learn how to multiply and divide negative numbers.
Multiplying When you multiply two or more numbers, you calculate their product. 2 × 3 equals 3 + 3, which equals 6. By the same logic, 2 × −3 equals −3 + −3, which equals −6. The number lines on the right show that −2 × 3 also equals −6.
When you multiply a positive number by a negative number or vice versa, the product is negative.
When you multiply two negative numbers, the product is positive.
Rules
positive × positive = positive + × + = +
positive × negative = negative + × = negative × positive = negative × + = negative × negative = positive × = +
examples
1 4 × 8 = 32 3 −4 × 8 = −32
2 4 × −8 = −32 4 −4 × −8 = 32
Dividing When you divide a number by another number, you usually check how many times the second number goes into the first number: 8 : 4 = 2, because 2 × 4 = 8. The same goes for negative numbers:
−8 : 4 = −2, because −2 × 4 = −8.
18 : −6 = −3, because −3 × −6 = 18.
−36 : −12 = 3, because 3 × −12 = −36. The result of a division is called the quotient
When you divide a positive number by a negative number or vice versa, the result is negative.
When you divide two negative numbers, the result is positive.
Rules
positive : positive = positive + : + = + positive : negative = negative + : = negative : positive = negative : + = negative : negative = positive : = +
examples
1 21 : 7 = 3 3 −21 : 7 = −3
2 21 : −7 = −3 4 −21 : −7 = 3
EXERCISES — PRACTICE
Multiplying
30 Copy and fill in positive or negative R
a When you multiply a negative number by a positive number, the result is
b When you multiply a number by a negative number, the result is positive.
c When you multiply a positive number by a …… number, the result is negative.
31 Read the explanation about multiplication on the previous page and take a look at the number lines. Perform the following multiplications using a number line. T1
a 2 × –3 d –1 × –3
b 1 × –3 e –2 × –3
c 0 × –3
32 Copy and fill in a minus sign if necessary. T1
a 2 × –15 = 30 e –8 × 9 = –72
b –13 × –4 = 52 f 7 × –6 = 42
c –9 × 2 = 18 g 10 × 3 = 30
d 10 × –3.5 = 35 h 2.5 × 5 = –12.5
33 Calculate. T1
a 2 × –3 g –10 × –10
b –6 × –4 h –15 × 0
c –3 × –5 i –2.5 × 6
d –7 × 1 j 5.25 × –20
e –3 × 12 k –7 × –1.5
f 5 × –14 l –50 × 0.5
Dividing
34 Copy and fill in the blanks. T1
a 28 : –4 = , because × –4 = 28.
b –64 : –8 = , because × –8 = –64.
c –35 : 5 = , because × 5 = –35.
d 27 : –3 = , because × –3 = 27.
35 Copy and fill in a minus sign if necessary. T1
a 45 : –5 = …9 e 72 : …24 = –3
b –14 : –2 = …7 f …36 : 6 = 6
c –56 : –7 = …8 g –100 : …25 = 4
d –25 : 10 = …2.5 h –40 : …8 = 5
36 Copy and fill in the blanks. T2
a –0.5 : 10 = …, because … × 10 = –0.5.
b 4.9 : –7 = …, because … × –7 = 4.9.
c –2 : –0.2 = …, because … × –0.2 = –2.
d 5 : –0.1 = , because × –0.1 = 5.
37 Calculate. Take another look at exercise 36 if necessary. T1
a –24 : 2 g –100 : –10
b 10 : –2 h –3 : –3
c –45 : –9 i 0 : –40
d 15 : –10 j 10 : –2.5
e –20 : 4 k –4 : 0.1
f 50 : –2 l –45 : –0.5
Order of operations It is important to perform calculations in the correct order. The correct order of operations is:
1 first perform all calculations inside brackets; 2 then multiply and divide from left to right; 3 lastly, add and subtract from left to right.
Make sure to perform the calculations step by step, writing down each step, as shown in the examples.
1
EXERCISES — PRACTICE
Order of operations
38 Write the following operations in the correct order: R
multiplication and division
addition and subtraction
calculations inside brackets
39 Calculate. Write down your calculations step by step, as shown in the example on the previous page. T1
a (20 + 4) : 6 e 18 : (15 – 6)
b 20 + 6 : 2 f 12 × 3 : 3
c 5 + 7 – 6 g 12 × 3 – 3
d 5 + (7 – 6) h (6 + 2) × 3
40 Calculate. Write down your calculations step by step, as shown in the example on the previous page. T2
a 2 × –3 × 4 e –10 : –10 : –10
b (40 + –13) : –3 f –4 + 8 × –4
c 160 : (–100 – –60) g 20 × –(7 + –9)
d –15 + 6 : –6 – –18 h –5 + –5 × 5 – –5
41 Copy and fill in the blanks. T2
a 3 × … = –12 e –2 × 3 + … = –12
b –75 : = –25 f 20 : –5 – = 3
c × –14 = 42 g –7 × + –1 = –50
d : –4 = 40 h : –6 – 4 = 6
42 You are babysitting one evening. Your call-out fee is 5 pounds and your hourly rate is 4 pounds. You are babysitting for three hours. How can you calculate your earnings? Write down your calculation. Also calculate how much you earn. T2
43 A cinema ticket costs ₤6.75. Making a reservation costs ₤0.50 per ticket. You make a reservation and buy 4 tickets. Which of the following calculations can you use to determine your costs? Also calculate how much you have to pay.
Note: there are multiple correct calculations. Explain your answer. T2
A 6.75 × 4 + 0.50 × 4
B 4 × 6.75 + 0.50
C (4 × 6.75) + (4 × 0.50)
D 4 × (6.75 + 0.50)
44 A handyman charges 32.50 pounds per hour and a call-out fee of 20 pounds. T2
a Write down a calculation that represents this situation.
b How much does the handyman charge for a job that takes five hours?
EXERCISES — EXPLORING & INVESTIGATING
exploring
45 In the United States, temperature is measured in degrees Fahrenheit. The temperature in degrees Fahrenheit can be converted to degrees Celsius by first subtracting 32, then multiplying the result by 5 and finally dividing it by 9:
°C = 5 × (°F – 32) : 9
A temperature of 50 degrees Fahrenheit corresponds to 5 × (50 − 32) : 9 = 5 × 18 : 9 = 90 : 9 = 10 degrees Celsius. T2
a Explain why ‘°F − 32’ is written inside brackets.
b The picture on the right shows a weather forecast for part of the United States. What is the expected temperature in Nashville in degrees Celsius? And how about Bismarck?
46 The following equations are incorrect because a pair of brackets is missing. Copy and insert the brackets in the correct spot. I
a –7 × 10 – –2 = –84
b 40 : –5 × 2 × –3 = 12
c –2.5 – 2.5 × 4 – 2 = –22
Weather forecast for the United States in degrees Fahrenheit (°F)
47 Copy and fill in the correct operations. Use +, , × or :. I
a –8 2 –5 = 1
b 10 –10 –10 = –110
c 5.25 –10 2.75 = 18
VOCAbuLArY
multiplying order of operations product brackets dividing number series quotient
Investigating
48 A number series is a series of numbers that follow a logical pattern. For example, in the series 1, 2, 4, 8, 16, every number is the previous number doubled. Number series are often part of psychological tests. Predict the next two numbers in the series shown below.
a –2, –4, –8, –16, –32, …, … T 1
b 15, 14, 12, 9, 5, … , … T 2
c 3, –6, 12, –24, 48, … , … T 2
d 1024, –512, 256, –128, … , … T 2
e 1, –1, 3, –5, 11, … , … I
49 Design three number series and ask one of your classmates to complete them. Include negative numbers as well as positive numbers. I
Have you reached your goal?
R I know the rules for multiplication and division with positive and negative numbers and I know the order of operations.
T1 I can multiply and divide positive and negative numbers.
T2 I can perform a calculation with multiple operations in the correct order.
I I can design my own number series with negative numbers.
COMP u TATIONAL s KILL s
C2 The Bay of Fundy, located on Canada’s east coast, has the largest tidal range in the world. Its tidal range (the difference between high and low tide) can be as large as 15 metres. The time between each high and low tide is about 6 hours.
I On average, how many centimetres does the water rise between low tide and high tide?
II On average, how many centimetres does the water rise every quarter of an hour between low tide and high tide?
High tide in the Bay of Fundy
Low tide in the Bay of Fundy
Fractions
GOAL You will learn what fractions are and how to add and subtract them.
Fractions When a number is divided by another number, the result may be a decimal number: 2 : 5 = 0.4. The division 2 : 5 can also be written as a fraction: 2 : 5 = 2 5, which is pronounced “two over five”, or, in a real-life situation, “two fifths (of a pie)”. The number above the line is called the numerator, the number below the line is called the denominator 2 5 = 2 : 5 = 0.4, so 2 5 = 0.4.
As fractions are numbers, they can be represented on a number line.
Negative fractions are written − 3 7, rather than − 3 7 or 3 − 7
A fraction can also be represented by the shaded area of a circle divided into equal parts. The improper fraction 17 5 is greater than 1 and can also be written as 325.
Fractions such as 32 5 and −21 7 are called mixed fractions.
simplifying Sometimes both the numerator and the denominator of a fraction can be divided by the same number. This is called simplifying the fraction. For example, both the numerator and the denominator of 4 6 can be divided by 2: 4 6 = 2 3.
Some fractions can be simplified to an integer: 10 10 = 1 and − 10 5 = −2. Always write fractions in their simplest form.
Adding and subtracting Fractions with the same denominator are called like fractions. As you can see by looking at the number line, two like fractions can be added or subtracted by adding or subtracting their numerators. Often, mixed fractions can easily be added or subtracted: 11 7 + 23
347. When you do not know the answer immediately, it might be helpful to write the mixed fractions as improper fractions before adding or subtracting them:
2 5 of the circle is shaded orange
EXERCISES — PRACTICE
Fractions
50 Which part of the figures is shaded orange? Give your answer as a fraction. T1
51 Draw a number line from 0 to 4 and indicate the location of the following fractions. T1
simplifying
54 a Use circles to show that 8 12 = 4 6 = 2 3 T1
b Use a number line to show that 4 10 = 2 5 T1
55 Write these fractions in their simplest form. T1 a 4 8 d 25 5 g –24 6
9 27 e 36 12 h 18 4
6 18 f –63 9 i –32 3
Adding and subtracting
56 Copy and fill in numerator or denominator. R
a In a fraction, the number below the line is called the …… .
b Like fractions are fractions that have the same
57 Calculate. T1
52 Write as mixed fractions. T1 a 11 2 d 21 7 g 37 4 b 15 4 e 8 3 h 32 5
c 20 9 f –27 8 i –29 6
53 Write as improper fractions. T1
a 13 4 d 55 6 g 40 1 7
b 2 5 6 e –71 3 h –4 3 7
c –15 9 f 25 1 4 i 10 3 8
4 5 –3 5
58 Calculate. T1
59 Copy and fill in the blanks. T2
FRACTIONS
Converting to like fractions When you convert fractions to like fractions you have to make sure their denominators are equal. This can be done as follows:
First multiply both the numerator and the denominator of the first fraction with the denominator of the second fraction.
Then multiply the numerator and the denominator of the second fraction with the numerator of the first fraction.
Sometimes, fractions can be converted to like fractions in an easier way. For example, when you want to convert 1 3 and 1 12 to like fractions, you do not need to convert both fractions so their denominator is 3 × 12 = 36. Instead, you can convert 1 3 to 1 × 4 3 × 4 = 4 12. Similarly, when you want to convert 5 6 and 2 9 to like fractions, you do not need to convert both fractions so their denominator is 6 × 9 = 54. Instead, you can use 18 as the denominator: 5 6 = 5 × 3 6 × 3 = 15 18 and 2 9 = 2 × 2 9 × 2 = 4 18.
examples
1 Convert 2 3 and 4 5 to like fractions.
2 3 = 2 × 5 3 × 5 = 10 15 =
2 Convert 1 2 and 3 8 to like fractions.
1 2 can be written as: 1 2 = 1 × 4 2 × 4 = 4 8
4 5 = 4 × 3 5 × 3 = 12 15 = Adding and subtracting unlike fractions Fractions with a different denominator are called unlike fractions. To add or subtract unlike fractions, you first have to convert them to like fractions.
examples
1
6, 12, 18, 24, ... 9, 18, 27, ...
3 Convert 1 6 and 5 8 to like fractions.
6, 12, 18, 24, 30, ... 8, 16, 24, 32, ... 1 6 = 1 × 4 6 × 4 = 4 24 5 8 = 5 × 3 8 × 3 = 15 24
EXERCISES — PRACTICE
Converting to like fractions
60 Copy and fill in the blanks. T1
a 1 4 = 12 d 4 7 = 12
b 2 5 = 10 e 5 = 8 20
c 2 3 = 12 f 3 = 15 40
61 Convert the following fractions to like fractions. T1
a 1 4 and 1 7 e 1 2 and 3 4 b 1 2 and 1 3 f 2 3 and 5 12
c 1 7 and 1 5 g 3 4 and 5 6 d 3 8 and 4 9 h 1 6 and 2 9
62 Copy and fill in < or >. T2
a 3 2 5 6 d 5 6 6 7
b 5 8 7 16 e 5 8 7 12
c 1 3 1 4 f 3 5 5 9
Adding and subtracting unlike fractions
63 Calculate. T1
a 1 4 + 1 3 e 2 3 + 4 5
b 2 5 + 3 7 f 1 6 − 1 4
c 3 4 − 1 8 g 5 9 + 7 10
d 5 6 − 3 5 h 2 7 − 5 9
64 Farmland in Groningen is used for several different purposes. About 3 7 of the land is used as grassland, 1 4 is used to grow grains, 3 20 is used to grow potatoes, 1 14 is used to grow sugar beets, 1 20 is used to grow cattle feed, and the remainder is used to grow other types of crops.
a Calculate the fraction of farmland that is used to grow potatoes or sugar beets. T1
b Calculate the fraction of farmland that is used to grow other types of crops. T2
65 Calculate. T1
3 1 4 13 8 g 75 6 211 12
51 6 + 23 4 h 47 8 31 4
66 Calculate. T1
a 25 6 − 2 3 e –1 2 + 31 2 b 5 1 11 61 4 f –2 – 32 3 c 61 8 +
EXERCISES — EXPLORING & INVESTIGATING
exploring
67 a Draw a number line from 0 to 1 and indicate the location of the fractions 2 5 and 3 5 Which fraction is larger? T1
b What part of a fraction is called the numerator? And what part is called the denominator? R
c How can you determine which of two like fractions is larger without using a number line? I
68 Calculate in an efficient way. I
a –50 6 7 – 2 3 7
b 47 1 13 – 49 2 13
c –115 2 3 + 215 1 6
Investigating 8 3 5 1 2 3
69 In a number pyramid, the number in every square is equal to the sum of the numbers in the two squares right below. An example is shown above. Copy the following number pyramid into your notebook and fill in the missing numbers. T2 2 5 6 4 5
70 Design your own number pyramid with four layers. Ask one of your classmates to complete the pyramid. I
bu LA r Y fraction simplifying numerator simplest form denominator like fractions improper fraction converting to like fractions mixed fraction unlike fractions
VOCA
COMP u TATIONAL s KILL s
C3 The total area of farmland in the Netherlands is 2 million hectares.
I Write 2 million as a number.
II Which number is a thousand times less than one million? Choose from the following: 100, 1000, 10 000, 100 000.
III Which number is a thousand times greater than one million? Choose from the following: one hundred thousand, one billion, one trillion.
IV One hectare is 10 000 m2. How many square metres of farmland are there in the Netherlands? Choose from the following: 200 million, 2 billion, 20 billion or 200 billion.
Have you reached your goal?
R I know what fractions are and what the numerator and denominator are. I also know the difference between like and unlike fractions.
T1 I can simplify fractions, convert them to like fractions, add them and subtract them.
T2 I can set up a calculation based on a word problem.
I I can design my own number pyramid with fractions.
Historical map of agricultural plots in the Schermer (North Holland), made by land surveyor Pieter Wils (1635).
1.4
Multiplying and dividing fractions
GOAL You will learn how to multiply and divide fractions.
Multiplying a fraction and an integer 4 × 2 5 and 2 5 × 4 can be written as 2 5 + 2 5 + 2 5 + 2 5. Therefore, both are equal to 4 ×
13 5
To multiply a fraction by an integer, multiply the fraction’s numerator by that number.
Note: When you multiply a mixed fraction, you first have to convert it to an improper fraction.
examples
Multiplying two fractions When you multiply 2 3 by 4 5, you are actually trying to find 2 3 of 4 5. You can calculate this as follows: 4 5 = 4 × 3 5 × 3 = 12 15 1 3 of 12 15 is 4 15. Therefore, 2 3 of 12 15 is
×
15 = 8 15. What you are actually doing is multiplying the numerators and denominators of both fractions: 2 3 × 4 5 = 2 × 4 3 × 5 = 8 15
To multiply two fractions, multiply their numerators and their denominators.
Note: When you multiply mixed fractions, you first have to convert them to improper fractions.
examples
×
× denominator
EXERCISES — PRACTICE
Multiplying a fraction and an integer
71 Calculate. T1
a 2 × 1 3 g 6 × 3 10
b 1 5 × 3 h 12 × 4 9
c 3 8 × 4 i –3 4 × 20
d 7 × − 5 7 j 6 × 1 3 4
e 3 7 × 2 k 2 1 7 × –14
f 2 3 × 5 l –3 1 5 × 7
72 Calculate. T1
a 1 3 of 12
b 7 10 of 40
c 3 8 of 56
d three quarters of 160
e two thirds of 120
f four fifths of 400
73 Calculate and fill in the blanks. T2
a 2 3 of a quarter of an hour is minutes.
b 1 12 of an hour is minutes.
c 1 100 of an hour is seconds.
d 1 20 of a century is years.
Multiplying two fractions
74 Calculate. T1
a 1 5 × 1 4 g 1 1 8 × 1 4
b 2 3 × 1 7 h 2 1 2 × − 3 5
c 5 6 × 2 9 i 2 1 2 × 3 5
d 7 8 × 8 7 j –1 4 × 1 3 8
–3 4
75 Calculate. T2
a –3 + 4 × 5 2
b (2 3 + 1 6) × 4 5 –2 5
c 2 (1 2 + 2 1 2) × 1 1 2
d 2 3 × 3 + (3 1 2 3)× 2
76 2 7 of a forest has been cut down. Of the newly available land, 3 4 is used for agriculture. What fraction of the original forest is now used for agriculture? T2
77 You went to bed at 10 pm last night and got up at 7 am this morning. You slept for three quarters of the time you were in bed. How long did you sleep? T2
MULTIPLYING AND DIVIDING FRACTIONS
Dividing a fraction by an integer 1 3 : 2 is half of 1 3 and is therefore equal to 1 3 × 1 2. This equals 1 3 × 2 = 1 6
To divide a fraction by an integer, multiply the fraction’s denominator by the integer.
Note: When you divide a mixed fraction, you first have to convert it to an improper fraction.
examples 1
Dividing a number by a fraction When you divide a number by a fraction, you determine how many times the fraction goes into the number: 4 : 2 5 = 10, because 10 × 2 5 = 4. Multiplying both sides by 5 2 gives 10 × 2 5 × 5 2 = 4 × 5 2. This means that 10 = 4 × 5 2, because 2
= 1. Therefore, you can find the result of 4 : 2 5 by multiplying 4 by the reciprocal of 2 5 The reciprocal of 2 5 is 5 2, so 4 : 2 5 = 4 × 5 2
To divide a number by a fraction, multiply the number by the reciprocal of the fraction.
Note: When you divide mixed fractions, you first have to convert them to improper fractions.
examples 1
EXERCISES — PRACTICE
Dividing a fraction by an integer
78 Calculate. T1
a 7 9 : 2 d 2 13 : 4
b 7 10 : 10 e 5 12 : 10
c 2 5 : 3 f 3 4 : 6
79 Calculate. T1
a 1 1 4 : 8 e 1 10 : 100
b 2 4 5 : 2 f 1 2 : 7
c 3 1 6 : 3 g 3 1 6 : 3 d 100 : 16 h 1 3 4 : 5
80 Copy and fill in the blanks. T2
a 3 4 : … = 1 4 c 24 27 : … = 1 9
b 8 3 : = 2 3 d –10 6 : = 12 3
81 4 5 of ₤27 500,– are divided among eight people. How many pounds does each person receive? T2
Dividing a number by a fraction
82 Calculate. T1
a 5 : 1 2 e 2 7 : 1 3
b 10 : 1 8 f 3 8 : 3 8
c 6 : 2 3 g 5 6 : 5 8
d 1 2 : 3 4 h 4 5 : − 3 2
83 Calculate. T1
: 5 6
d 11 5 : 11 5 h 3 1 3 : 21 2
84 Calculate. T2
a 2 : (2 7 + 1 14)
b ( 3 10 + 1 2 5) : 7 10
c –6 : 1 3 – 2 × 5 3
d 1 2 × (1 3 4 – –1 1 2) : 1 2
85 In 2016, Dutch people drank 40 litres of milk on average. One glass of milk contains about 1 5 of a litre. How many glasses of milk did the average Dutch person drink in 2016? T2
The IJsselkop near Arnhem. The Pannerden Canal splits into the Lower Rhine (left) and the IJssel (right).
exploring
86 The Rhine crosses the Dutch border near Spijk, a village in Gelderland. On average, 2.1 million litres of water flow down the Rhine each second. At the Pannerdense Kop, the Rhine splits into the Waal and the Pannerden Canal. The Directorate-General for Public Works and Water Management (Rijkswaterstaat) determines how much water flows down each river. Under normal circumstances, 2 3 of the water from the Rhine flows down the Waal and 1 3 flows down the Pannerden Canal.
a Calculate how much water, on average, flows from the Rhine into the Waal each second. T2
Near Arnhem, the Pannerden Canal splits into the Lower Rhine and the IJssel. Of all the water that flows down the Pannerden Canal, 2 3 ends up in the Lower Rhine and 1 3 ends up in the IJssel.
b Calculate how much water, on average, flows down the Lower Rhine each second. Write your answer as a fraction if necessary. I
IJsselkop
Lower Rhine Spijk
Pannerdense Kop
Waal
Rhine IJssel
COMP u TATIONAL s KILL s
C4 Take another look at exercise 86.
I Estimate how many billions of litres of water flow down the Rhine each day.
II Now calculate how many billions of litres of water flow down the Rhine each day.
87 Write down three multiplications that have a result of 3 5 I
88 a Copy and finish the calculation. T1
11 4 × 8 19 = 11 × 4 × =
b When you multiply 11 4 by 8 19, the numerator and denominator contain a common factor 4. Copy and finish the calculation. T1
11 4 × 8 19 = 11 4 × 4 × 2 19 = 11 1 × 2 19 = …
c Calculate 9 25 × 150 7 , 300 13 × 7 15 and 39 125 × 625 26 . T2
d Are you allowed to divide out the common factor 4 in the division 11 4 : 8 19 = 11 4 : 4 × 2 19 ?
Explain. I
89 Copy and fill in the blanks. I
a 1 11 × … = 1 33 e –1 7 × … = 11 7
b 2 3 : = 1 f 11 3 × = 1
c 4 5 × = –12 25 g 3 8 : = –9 4
d –3 10 : = –4 9 h – 11 3 : = –51 3
Investigating
90 Together with a classmate, make a video or a presentation in which you explain either how to multiply or how to divide two fractions. I
VOCA bu LA r Y
reciprocal
Have you reached your goal?
R I know the rules for multiplying and dividing fractions.
T1 I can perform multiplications and divisions with fractions.
T2 I can set up a calculation based on a word problem.
I I can explain why dividing out a common factor or changing the order of operations is allowed in multiplications, but not in divisions.
GOAL You will learn what powers are and how to perform calculations with them.
Powers and raising to a power Multiplying a number by itself several times is called raising that number to a power. For example, 2 × 2 × 2 is 2 raised to the power of three. 2 × 2 × 2 can be written as 23 and 6 × 6 × 6 × 6 can be written as 64. 64 can be pronounced in more than one way: ‘six raised to the fourth power’, ‘six to the fourth’, or ‘six to the power of four’. When raising a negative number to a power, the negative number must be enclosed by brackets. For example, (−2)4 = 16. −24 (no brackets) is equal to the opposite of 24, which is −16.
examples
1 62 = 6 × 6 = 36
2 (− 6)2 = − 6 × − 6 = 36
3 − 62 = − 6 × 6 = − 36
4 (1 2)4 =
5 (11 2)3 = 1 1 2 × 1 1 2 × 1 1 2 = 3 2 ×
base and exponent The power 23 consists of base 2 and exponent 3. The exponent is the small number written above and to the right of the base: base exponent. In 23 the exponent 3 determines what happens to the base 2.
squaring Raising a number to the second power is very common and therefore has a special name: squaring. 5 × 5 can be written as 52, which is pronounced 5 squared. The word squared comes from the Latin word ‘exquadrare’, which means ‘to square’.
Knowing the squares of the numbers 1 through 15 by heart will make it easier to perform calculations.
12 = 1 62 = 36 112 = 121
22 = 4 72 = 49 122 = 144
32 = 9 82 = 64 132 = 169
42 = 16 92 = 81 142 = 196
52 = 25 102 = 100 152 = 225
These are additional useful squares to remember:
202 = 400 302 = 900 502 = 2500
252 = 625 402 = 1600 1002 = 10 000 2 3 base exponent 4
4 The area of a square is equal to the length of its side squared: area = 4 × 4 = 42 = 16
EXERCISES — PRACTICE
Powers and raising to a power
91 Copy and fill in the blanks. T1
a 33 = 3 × 3 × 3 =
b 52 = 5 × 5 =
c 24 = − 2 × 2 × 2 × 2 =
d ( 3)3 = − 3 × − 3 × − 3 =
e 18 = …
f ( 4)2 = …
92 Copy and fill in the blanks. T1
a (1 5)3 = 1 5 × 1 5 × 1 5 = …
b (11 2)2 = 1 1 2 × 1 1 2 = …
c (2 5)2 = …
d (21 4)2 = 2 1 4 × 2 1 4 = 4 × 4 = …
e (11 3)3 = …
f (41 2)2 = …
93 Calculate. T1
a (0.5)2 e (– 4 )4 i (3 4)3
b 162 f 100 3 j 106
c 1000 2 g –36 k (21 7)2
d 312 h (1 3)2 l (12 3)3
94 Copy and fill in the blanks. T1
a –1 × –1 = (–1) c 9 × 9 × 9 × 9 = 4
b 2 5 × 2 5 × 2 5 = (2 5) d 7 9 × 7 9 = 2
base and exponent
95 Determine the base and the exponent of the powers. T1
a 5 10 c 342 e (51 3)10
b 214 d (1 2)8 f (41 2)2
96 Determine the base of the powers. T2
a 144 = … 2 c 81 = … 4
b 125 = … 3 d 1 000 000 000 = … 3
97 Determine the exponents. T2
a 27 = 3 c 1 16 = (1 4)
b 625 = 5 d 1 32 = (1 2)
98 a Calculate (−2)2, (−2)3, (−2)4 and (−2)5 T1
b Calculate (−3)2, and (−3)3 T1
c When does a power with a negative base have a positive result? I
squaring
99 Calculate. R
a 112 c 152 e 302 b 142 d 252 f 402
100 Two squares are shown below. Write their areas as a square. T1
Order of operations It is important to perform calculations in the correct order. If a calculation contains powers, the powers have priority over multiplications, divisions, additions and subtractions. Brackets have priority over powers, so calculations inside brackets have to be performed first. The correct order of operations is:
1 first perform any calculations inside brackets; 2 then calculate any powers or squares; 3 then multiply and divide from left to right; 4 lastly, add and subtract from left to right.
Make sure to perform the calculations step by step, writing down each step, as shown in the examples below.
EXERCISES — PRACTICE
Order of operations
101 Write the following operations in the correct order. R
raising to a power
calculations inside brackets
addition and subtraction
multiplication and division
102 Copy and fill in the blanks. T1
a 32 + 52 = … + … = …
b (9 + 2)2 = … 2 = …
c 5 × 24 = 5 × … = …
103 Calculate. T1
a 22 + 142
b 6 × 52
c (21 + 9) × 10
d 100 − 53
e 16 : ( 2)3
f 52 × − 32
g 3 + 2 × 72
h 202 : (8 × 5)
104 Calculate. T1
a ( 4 × 2 + 10)3
b 62 × 3 – 6 × 32
c (22 – 7) : 3 + 6
d 2 × (142 – 42)
e (1 + 2) 3 + 4 × 5
f (5 + 2) × (1 42)2
105 Calculate. T2
a 4 × (1 2)3
b (2 3 − 3 7)2
c 1 + 2 × ( 5 7)3
d 24 : 1 1 2 1
e (1 3 + 7 4) × 22 + 3
f 42 × (–3 2 + 4)
106 Calculate. T2
a 1 5 × (53 – 52)
b 1 1 3 + (21 3)2
c 5 : 102 : 1 5
d (1 4 –1 3) : 1 24
e –22 × 23 : (1 2)2 – 2
f 31 5 : 33
107 The following calculations are incorrect, because a pair of brackets is missing. Copy the calculations and include brackets in the correct places. I
a 3 + 5 × 22 = 32
b 10 3 × 23 + 5 = − 29
c 4 × 23 22 × 3 = 48
d 22 × 3 2 × 32 = − 24
exploring
108 a How is 52 pronounced? R
b A room has an area of 16 m2 How is ‘m2’ pronounced in this case? T2
c Which exponent is used to refer to an area? T2
d Which exponent is used to refer to a volume? T2
109 The power 23 is pronounced ‘two cubed.’ Explain the relationship between a cube and a third power. I
110 a Explain why 1 dm2 is equal to 100 cm2. T2
b Explain why 1 dm3 is equal to 1000 cm3. T2
c What calculation do you need to perform to determine how many mm3 go into 1 dm3? Write down your calculation using a base and an exponent. T2
111 In the figure below, each branch splits into three new branches. T2
a The figure consists of 1 + 3 + 32 + 33 branches. Explain why and calculate the total number of branches.
b How many branches are added if you draw the next level of offshoots?
Investigating
112 Read the text Grains and a chessboard on the right page.
a How many grains are placed on squares 5 through 10? T1
b Write your answers to part a as powers of 2. T2
King Shirham asked his servants to get him a chessboard. Then he started to put the right number of grains on each of the squares. At first, this seemed like an easy task. However, he ran into a problem pretty quickly, because 16 777 216 grains had to be put on square 25.
c Write down the number of grains on square 25 as a power of 2. T2
d Calculate how many grains need to be placed on square 26. T2
e Write down the number of grains on square 64 as a power of 2. T2
f Do you think King Shirham had enough grains to be able to give Sissa his reward? Explain your answer. I
COMP u TATIONAL s KILL s
C5 Calculate. I 412 – 67 – 12
45 – 12.31 – 0.69
5.1 – 3.7 – 1.19
735 : 15
5.115 : 11
Grains and a chessboard
King Shirham was a tyrant. One of his advisors, Sissa, invented the game of chess to show the King he needed to take care of all the people in his country, because they were all equally important.
The King was very enthusiastic about the game: chess taught him that the farmers (pawns) and the nobility (the other chess pieces) needed to work together as one unit.
King Shirham promised Sissa to give him a reward, anything he wanted. Sissa then asked the King to place one grain on the first square of a chessboard, two grains on the second square, four grains on the third square, eight grains on the fourth square, etc. Each square had twice as many grains as the previous square. What a stupid man, thought the King. I would have given him a much larger award.
IBN KHALLIKAN (2015). Biographical dictionary of Ibn Khallikan, vol. 3. Arkose Press.
bu LA r Y raising to a power base exponent the square of squaring order of operations
Have you reached your goal?
R I know what powers, squares, bases and exponents are and I know the order of operations.
T1 I can raise a number to a power and I can perform calculations in the correct order.
T2 I can determine the base and the exponent in calculations that include powers.
I I can explain the relationship between powers and areas, volumes and the story about grains on a chessboard.
VOCA
Test preparation
At the end of every section, check whether you understand the vocabulary and have reached your learning goal. If not, reread the explanation or take another look at the videos. Then do the following exercises.
Mathematics in daily life
113 a Write 19 as a binary number and as a byte. T1
b Write the binary number 100101 as a decimal numeral. T1
114 Which byte has the largest value? Give the value of this byte as a decimal numeral. I
1.2
117 Copy and fill in positive or negative R
a positive × negative =
b negative × = positive
c : positive = negative
d negative : negative =
118 Copy and fill in the blanks. T2
1.1
115 Calculate. T1
116 Alexander the Great was the King of Macedonia and, due to his many conquests, ruled over one of the largest empires in history. His empire stretched from Greece to the Himalaya Mountains. Alexander the Great was born on 20 July in 356 BC and died on 11 June in 323 BC. Calculate how old he was when he died. T2
Alexander the Great
119 Calculate. T1
a 10 – 4 + 2 × 6
b 5 + 8 – 16
c 6 × 5 – 8 : 2
d (2 – 5) – (3 – 4)
1.3
120 Copy and fill in numerator or denominator R
a Fractions that have the same are called like fractions.
b Fractions with the same can be added by adding their
121 Calculate. T1
1.4
122 Calculate. T1
a 3 5 × 7 e 17 × 33 34
b 4 7 × − 5 6 f 1 3 × 12 5
c 3 8 : 2 g −4 : − 1 3
d 2 11 : 3 4 h 12 3 : 13 5
123 About 1 4 of the farmland in the province of Groningen is used to grow grains. Of this land, 4 5 is used to grow wheat and 1 5 is used to grow barley. T2
a Calculate what part of farmland in Groningen is used to grow wheat and what part is used to grow barley. Write your answers as a fraction.
b In 2017, 28 thousand hectares of farmland in Groningen were used to grow wheat. Calculate the number of hectares of farmland used to grow barley.
1.5
124 Which of the numbers in a power is the base and which is the exponent? R
125 Calculate. T2
a 4 + (23 : 4)2
b (11 9 –2 9) × (11 2 )2
c (6 + 4) : (22 + 1)
d 3 − (22 + 32)2 + 5
Chapter 1
126 Copy and fill in <, > or = . T2 a 1 … −1 e 1 3 … 1 4 b 19 20 f 3 5 2 3
c 5.15 5.07 g 3 5 0.60
d 0.1 0.09 h 5 7 11 14
127 The following calculations are incorrect, because a pair of brackets is missing. Copy the calculations and include brackets in the correct places. T2
a 3 − 4 × 2 + 5 = −25
b 1 4 + 1 2 × 2 3 1 6 = 1 2
c 1 4 × 23 22 × 3 = 3
128 Calculate (−1)1000 I