METHOD CONCEPT / EDITOR
Boom voortgezet onderwijs
AUTHORS
Benjamin del Canho
Maartje Elsinga
Jacqueline Kooiman
Gijs Langenkamp
Chantal Neijenhuis
Sibren Stienstra
Roosmarij Vanhommerig
Vera de Visser
KERN WISKUNDE
HAVO
YEAR 3 PART A
BOOM VOORTGEZET ONDERWIJS
© 2022 Boom voortgezet onderwijs, Groningen, The Netherlands
Behoudens de in of krachtens de Auteurswet van 1912 gestelde uitzonderingen mag niets uit deze uitgave worden verveelvoudigd, opgeslagen in een geautomatiseerd gegevensbestand, of openbaar gemaakt, in enige vorm of op enige wijze, hetzij elek tronisch, mechanisch door fotokopieën, opnamen of enig andere manier, zonder voorafgaande schriftelijke toestemming van de uitgever.
Voor zover het maken van kopieën uit deze uitgave is toegestaan op grond van artikelen 16h t /m 16m Auteurswet 1912 jo. besluit van 27 november 2002, Stb 575, dient men de daarvoor wettelijk verschuldigde vergoeding te voldoen aan de Stichting Reprorecht te Hoofddorp (postbus 3060, 2130 kb , www.reprorecht.nl) of contact op te nemen met de uitgever voor het treffen van een rechtstreekse regeling in de zin van art. 16l, vijfde lid, Auteurswet 1912. Voor het overnemen van gedeelte(n) uit deze uitgave in bloemlezingen, readers en andere compilatiewerken (artikel 16, Auteurswet 1912) kan men zich wenden tot de Stichting PRO (Stichting Publicatie- en Reproductierechten, postbus 3060, 2130 kb Hoofddorp, www.stichting- pro.nl).
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 6442 003 6 www.boomvoortgezetonderwijs.nl
KERN Wiskunde 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 Luuk Wilhelm
Book design & cover René van der Vooren, Amsterdam
Layout & technical drawings PPMP, Wolvega
1 Numbers and variables
mathematics in daily life
Gearwheels 8
1.1 Types of numbers 10
1.2 Roots 16
1.3 Powers 22
1.4 Expanding brackets 28
1.5 Fractions with variables 34
Test preparation 40
2 Geometric reasoning
mathematics in daily life
Aspect ratio of photographs 44
2.1 Similarity 46
2.2 Similarity postulates 52
2.3 Angle bisectors and perpendicular bisectors 58
2.4 Altitudes and medians 64
2.5 Inscribed and circumscribed circles 70
Test preparation 76
3 Relations
mathematics in daily life Why are urban areas warmer than rural areas? 80
3.1 Linear relationships 82
3.2 Quadratic relationships 88
3.3 Different types of relationships 94
3.4 Translating, stretching and compressing graphs 100
3.5 Periodic relationships 106
Test preparation 112
4 Statistics
mathematics in daily life Florence Nightingale, the lady with the lamp 116
4.1 Tables 118
4.2 Graphs and charts 124
4.3
4.4
Grouping data 130
Measures of central tendency 136
4.5 Measures of spread 142
Test preparation 148
Appendix 150
Index 162
(a + b) (c + d) = ac + ad + bc + bd
1
Numbers and variables
In this chapter you will learn about different types of numbers and their properties. You will also learn how to perform calculations with roots and powers, how to expand brackets and how to perform calculations with fractions that contain variables. In addition, you will be introduced to gearwheels, bicycle gears and the Lichtenberg Ratio. Finally, you will learn about distances in our solar system and about the relation between Pythagorean tuning and music.
MATHEMATICS IN DAILY LIFE
Gearwheels 8
1.1 Types of numbers 10
1.2 Roots 16
1.3 Powers 22
1.4 Expanding brackets 28
1.5 Fractions with variables 34
Test preparation 40
MATHEMATICS IN DAILY LIFE
GOAL You will learn how gearwheels transmit movement and what a bicycle’s gear inches are.
Gearwheels
Gearwheels Gearwheels help transmit movement. Below you see an example of two intermeshing gearwheels.
When the left gearwheel starts rotating in one direction, the right gearwheel will rotate in the opposite direction. The teeth or cogs on both gearwheels mesh one by one. The two gearwheels do not have the same number of teeth. Therefore, they do not rotate at the same speed. The smaller gearwheel has 20 teeth, while the larger gearwheel has 30 teeth. If you want the larger gearwheel to make one complete turn, the smaller gearwheel needs to turn 30 20 = 1.5 times. This means that the smaller gearwheel rotates faster than the larger one.
The ratio 30 20 = 1.5 is called the gear ratio of the intermeshing gearwheels. A ratio greater than 1 enables you to slow down the movement. A ratio less than 1 enables you to speed it up.
b icycle’s gear inches While cycling, you push the pedals of a bicycle, which causes the back wheel to rotate in the same direction, because a chain connects a gearwheel on the bottom bracket axle to a gearwheel on the rear axle.
chain
rear axle bottom bracket axle
When the gearwheel attached to the bottom bracket axle has 45 teeth and the gearwheel on the rear axle has 15 teeth, then the back wheel will rotate 45 15 = 3 times every time the pedals make one complete turn. If the diameter of the back wheel is 28 inches, you will cover a distance of 3 · 28 · π ≈ 264 inches when the pedals rotate once. This distance is called the bicycle’s gear inches
A bicycle with gears has multiple gearwheels on its rear axle, and sometimes it also has multiple gearwheels on its bottom bracket axle. You can change gears while cycling. Using a higher gear means that you can reach a higher speed, but pedalling will be harder. A lower gear requires less effort. This is convenient when cycling uphill or into a headwind.
In a mechanical watch many small gearwheels mesh into each other. This makes the hour hand and the minute hand rotate at the correct speed.
EXERCISES
Questions about the text
1 Two gearwheels mesh into each other. The smaller gearwheel has 20 teeth, the larger gearwheel has 32 teeth.
a Calculate the gear ratio of the intermeshing gearwheels. T1
b How many times does the larger gearwheel rotate when the smaller one rotates 4 times? T1
c Both gearwheels have made a complete number of rotations. What is the minimum number of rotations that the smaller gearwheel has made? I
2 A bicycle has a gearwheel with 52 teeth on the bottom bracket axle. The gearwheel on the rear axle of the bicycle has 16 teeth. The diameter of the back wheel is 26 inches.
a Calculate the bicycles’ gear inches. Round your answer to the nearest inch. T1
b There are two more gearwheels on the rear axle, one with 14 teeth and one with 18 teeth. You want to increase your gear inches. To which gearwheel should you shift the chain? Explain your answer. T2
Have you reached your goal?
R I know what the gear ratio of two intermeshing gearwheels is and I know what a bicycle’s gear inches are.
T1 I can calculate the gear ratio of two intermeshing gearwheels and I can calculate a bicycle’s gear inches.
T2 I know in which direction gearwheels rotate when there are more than two gearwheels, and I can calculate the gear ratio of those gearwheels.
I I can explain whether several intermeshing gearwheels can move.
in–depth question
3 Four gearwheels that mesh into each other are shown below.
a If gearwheel 1 moves in an anticlockwise direction, what will be the direction of gearwheel 4? T2
b Gearwheel 1 rotates five times. How many times do the other gearwheels rotate? T2
c Calculate the gear ratio of gearwheel 1 to gearwheel 4. Suppose gearwheel 1 and gearwheel 4 were directly connected. What would the gear ratio be? What do you notice? T2
d What problem would occur if a fifth gearwheel were connected to gearwheels 1 and 4? I
Research assignment
4 Find three different kinds of bicycles in a bicycle parking or on the internet. Ensure that only one of the bicycles does not have gears. Calculate all gear inches of these three bicycles and compare the distances to each other. Present your findings in a presentation, video, poster, quiz or an advertorial. T2
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gear ratio
gear inches
Types of numbers
GOAL You will learn about different types of numbers. You will also learn how to find the prime factorisation of a number.
Natural numbers and integers The numbers 0, 1, 2, 3, ... are called natural numbers*. The dots indicate that this sequence of numbers is infinite. The numbers ..., –3, –2, –1, 0, 1, 2, 3, ... are called integers. For example, –8 and 2 are integers. 2 is also a natural number, but –8 is not.
Examples
1 5 is an integer and a natural number.
2 –100 is an integer, but not a natural number.
3 The result of (–1)4 is a natural number, because (–1)4 = 1.
4 The result of (–1)5 is an integer, but not a natural number, because (–1)5 = –1.
5 11 2 is not an integer, nor is it a natural number, because 11 2 = 1.5.
Prime numbers An integer greater than one with exactly two positive divisors, 1 and itself, is called a prime number. There is an infinite number of prime numbers. The first five prime numbers are 2, 3, 5, 7 and 11. Every integer greater than 1 can be written as a product of prime numbers. For instance, 15 can be written as 3 · 5. Finding these prime numbers is called prime factorisation. In this case, 3 and 5 are the prime factors. When a number is a prime number, there is only one prime factor: the number itself. You can find a prime factorisation by systematically finding larger prime factors of a number. An example for the number 60 is shown on the right.
Examples Give
factorisation of the numbers 40 and 99.
* Sometimes 0 is excluded from the set of natural numbers. In this book 0 is always included.
EXERCISES — PRACTICE
Natural numbers and integers
5 a What is a natural number? R
b Draw a number line from 0 to 10 and indicate the location of the natural numbers. T1
6 Which of the following numbers are natural numbers? And which are integers? T1
7 √16 99 5 7 √8 100 25 0 1 2 9 0.8 5000 ( 1)8 104 ( 2)3 ( 6)101
7 For each of the following divisions, indicate whether its quotient is a natural number. T1
a 55 : 8 c –0.1 : 0.01
b 200 : 0.5 d –125 : –25
8 Even numbers can be written in the form 2n and odd numbers can be written in the form 2n + 1, where n is an integer. Write the following numbers in one of these forms. T2
a 34 c 113 b 87 d 228
Prime numbers
9 a What is a prime number? R
b Give the first five prime numbers. R
c Why is 25 not a prime number? T1
d Give all prime numbers between 20 and 30. T1
10 In this exercise you will find the prime factorisation of 150. T1
a Copy and fill in the diagram shown below.
b Give the prime factorisation of 150.
11 Give the prime factorisation of the following numbers. T1
a 16 c 66
b 40 d 210
T Y p ES OF N u M b E r S
Rational, irrational and real numbers When you divide 2 by 5 you get 0.4. Therefore, the number 0.4 can be written as the fraction 2 5. In a fraction, the numerator and denominator are integers. The denominator cannot be 0, because division by 0 is undefined. Numbers that can be written as a fraction are called rational numbers. There are many rational numbers. For example, 0.75, –0.666666... and 11 3 are rational numbers, because 0.75 = 3 4, –0.666666 = –2 3 and 11 3 = 4 3
Some numbers cannot be written as a fraction. These numbers are called irrational numbers. For example, √2 = 1.414213... and π = 3.141592... cannot be written as a fraction. There is no division whose quotient is √2 or π. There are many irrational numbers. For example, the square root of a natural number that is not a perfect square is always an irrational number. Together, the rational and irrational numbers are called the real numbers. Every number on a number line is a real number.
Rational numbers are numbers that can be written as a fraction. Irrational numbers are numbers that cannot be written as a fraction.
The orange numbers are irrational numbers.
Examples
1 0.125 is a rational number, because 0.125 = 1 8
2 8.231 is a rational number, because 8.231 = 8231 1000
3 2 is a rational number, because 2 = 2 1
4 –61 7 is a rational number, because –61 7 = 43 7
5 √41 = 6.403124… is an irrational number.
6 √1 4 = 1 2, so √1 4 is a rational number.
Rational, irrational and real numbers
12 Copy and fill in rational or irrational R
a A number that can be written as a fraction is a(n) number.
b A number that cannot be written as a fraction is a(n) number.
13 a What is the definition of a real number? R
b Draw a number line from –1 to 2 and use arrows to indicate five different rational numbers. T1
c Use arrows to approximately indicate three irrational numbers on the same number line. Write the numbers next to the arrows. T1
14 For each of the numbers below, explain whether it is a rational or an irrational number. T1
15 Copy the table. In each of the cells, put a tick if the number belongs to the set on the left and a cross if the number does not belong to the set on the left. T1
1 natural number integer rational number irrational number real number
16 For each of the following square roots, explain whether it is a rational or an irrational number. T2
Exploring
17
a Give the prime factorisation of 84. T1
b Determine the number of positive divisors of 84 by using its prime factorisation. T2
18 When you write a fraction as a decimal number, the number of decimals is finite if the denominator is a power of 10 or if the denominator can be converted to a power of 10.
For example, 345 1000 = 0.345 and 11 8 = 11 125 8 125 = 1375 1000 = 1.375.
Write the following decimal numbers as fractions without using a calculator. T2
a 3 25 c 339 500 b 9 200 d 199 1250
19 Suppose you can only jump 33 units to the right or 75 units to the left on a number line. You start at 0. The number line below shows that you will end up at –9 after two jumps to the right and one to the left. I
investigating
20 Read the text Repeating decimals on the next page.
a Write the fractions 4 15 and 10 37 as decimal numbers. T1
b Give the first ten decimals of 0.828 and 0.125160. T1
c Give the hundredth decimal of the numbers in part b. T2
21 Consider the number A = 0.18.
a Show that 100A – A = 18. T2
b Show that from this, it follows that A = 2 11. T2
c Write 0.225 as a fraction. I
d Write 0.9 as a fraction. I
22 a √2 is an irrational number. Explain why √2 + 1 is also an irrational number. I
b π is an irrational number. Explain why 2π is also an irrational number. I
a You start at 0 and jump three times to the left and twice to the right. At which number do you end up?
b You start at 0 and jump five times to the left. What is the minimum number of times you have to jump to the right to end up at a natural number? Which number is this?
c Explain whether you can end up at any possible integer if you start at 0.
CO m P u TaT i ON al s K ills
C1 There are 7351 rooms in the First World Hotel in Pahang, Malaysia.
i Round the number of rooms to the nearest hundred.
ii Round the number of rooms to the nearest thousand.
iii Another hotel has 4700 rooms, rounded to the nearest hundred. So there are fewer rooms in this hotel than in the First World Hotel. How many fewer? Give the minimum as well as the maximum number.
Repeating decimals
When you write a fraction as a decimal number, the number of decimals may be infinite.
For example, 1 3 = 0.3333333… and 3 11 = 0.272727… . Here the decimals repeat themselves forever. Therefore, these decimal numbers are called repeating decimals. To make these decimal numbers easier to write, a line or dot can be placed over the repeating pattern: 1 3 = 0 . 3 and 3 11 = 0 . 27. Note: the repeating pattern does not always start directly after the decimal point. For example, 5 6 = 0.8333333… = 0.83.
Every fraction written as a decimal number (and therefore every rational number) either has a finite number of decimals or an infinite number of decimals with a repeating pattern. The number of decimals can be zero (for instance, 7 1 = 7).
In reverse, any decimal number that has an infinitely repeating pattern after the decimal point can be written as a fraction. This means that irrational numbers never have an infinitely repeating pattern after the decimal point.
Have you reached your goal?
R I know what natural numbers, integers, rational numbers, irrational numbers and real numbers are. I also know what prime numbers are.
T1 I can identify types of numbers and I can find the prime factorisation of a natural number.
T2 I can write a fraction as a decimal number by converting the fraction’s denominator to a power of 10.
I I can think about and give reasoned responses to questions about different types of numbers.
natural number integer prime number prime factorisation prime factor rational number irrational number real number repeating decimal
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Roots
GOAL You will learn how to perform calculations with square roots. You will also learn what nth roots are and how to perform calculations with them.
Performing calculations with square roots The square root of a number a is equal to the nonnegative number whose square is a For example, √36 = 6, because 62 = 36. The square root is also called the second root or root of degree 2. Taking a square root is the opposite of squaring. Like radical terms are terms with the same radicals. Like radical terms can be added or subtracted. For example,
2 √3 + 4 √3 = 6 √3 . Square roots can also be multiplied or divided.
The following rules apply: √a √b = √ab and √a √b = √a b .
Examples
1
2
Examples
1
simplifying square roots Some square roots can be simplified. When the radicand is divisible by a perfect square, you can take out the square root of that perfect square and place it in front of the radical. Use the rules for multiplying square roots. Always simplify as much as possible.
Examples 1
Performing calculations with square roots
23 Calculate without using a calculator. T1
a √49 c √144
b √81 d √400
24 Copy and fill in the blanks. T1
27 Calculate without using a calculator. T1 a √2 √5 d √15 √3 b 2 √5 8 √7 e 9 √14 3 √2 c 6 √3 3 √3 f 4 √8 √2
simplifying square roots
28 Simplify. Do not use a calculator. T1
25 a What is the length of the sides of a square with an area of 196 cm 2? T2
b What is the length of the sides of a square with an area of 200 cm 2? Round your answer to the nearest hundredth of a cm. T2
26 Calculate without using a calculator. T1
29 Calculate the lengths of sides AC and EF of the two triangles shown below. Simplify as much as possible. T2
n th roots It is also possible to find the nth root of a number. The n th root is also called the root of degree n. The n th root of a number raised to the power of n equals that number. For example, if you want to determine the length of the edges of a cube with a volume of 8, you want to find the number that equals 8 when raised to the power of three. This number is the third root, or cube root, of 8 and is written as 3√8 . Since 23 = 8, 3√8 = 2.
The n in n th root can be any number. For example, 4√81 is the number that equals 81 when raised to the power of four. 3 4 = 81, so 4√81 = 3.
Make sure to note the following when working with n th roots:
It is not possible to extract an even root from a negative number. For instance, 4√ 16 does not exist, because a number raised to the power of four is always greater than or equal to 0.
It is possible to extract an odd root from a negative number. For instance, 3√ 8 = –2, because (–2)3 = 8.
Even roots are always positive. For instance, 4√81 = 3, even though 3 4 = 81 and (–3) 4 = 81. Due to this agreement, it is clear which number 4√81 equals.
Examples
1 3√27 = 3, because 33 = 27 4 5√ 1 32 = 1 2, because (1 2)5 = 1 32
2 4√256 = 4, because 44 = 256 5 5√ 1 = − 1, because (− 1)
3 3√ 216 = − 6, because (− 6)3 = − 216 6 6√ 1 does not exist
approximating n th roots Many square roots cannot be written as a fraction and are therefore irrational. This also applies to many n th roots, such as 3√100 . The number 3√100 is greater than 33√64 = 4 and less than 3√125 = 5. Therefore, 4 < 3√100 < 5. You can find a more accurate approximation using your calculator: 3√100 = 4.6415..., so 3√100 ≈ 4.64.
EXERCISES — PRACTICE
nth roots
30 Copy and fill in the blanks. T1
a 53 = , so 3√125 =
b 24 = , so 4√16 =
c (−4)3 = , so 3√ 64 =
d 15 = …, so 5√1 = …
31 Copy and fill in the blanks. T1
a 3√27 = 3, because … 3 = …
b 4√625 = …, because … 4 = …
c 3√ 8 = …, because … 3 = …
32 Copy and fill in the blanks. T1
a 3√1 8 = 1 2, because … 3 = …
b 4√ 1 81 = , because 4 =
c 3√ 8 125 = , because 3 =
33 (–5) 4 = 625. Explain why 4√625 ≠ –5? T2
34 Explain why 3√ 1 exists, but 4√ 1 does not. T2
35 Calculate without using a calculator. T1
a 4√81 d 3√1 8
b 5√32 e 3√0.001
c 3√−125 f 999√−1
approximating nth roots
36 Calculate. Round your answer to the nearest thousandth. T1
a 3√100 d 3√−35
b 4√2 e 10√1002
c 7√−98 f 3√6.89
37 Third roots can be approximated without using a calculator. T1
a Copy the table and fill in the missing values. number 0 1
b The number 50 lies between the cubes 27 and 64. Between which two integers does 3√50 lie?
c Between which two cubes does 300 lie?
d Between which two integers does 3√300 lie?
38 What is the length of the edges of a cube with a volume of 216 cm 3? T1
Exploring
39 Calculate if possible. If not possible, explain why. Do not use a calculator. T2
a
40 Calculate and indicate whether the result is a rational or an irrational number. Do not use a calculator. T2
investigating
42 Read the text Paper sizes on the next page. T2
a The dimensions of an A0 sheet of paper are shown on the next page. Show that these dimensions conform to the Lichtenberg Ratio.
b Calculate the dimensions of an A5 sheet of paper.
c Calculate the area of an A11 stamp. Round your answer to the nearest cm 2 .
43 A sheet of paper conforms to the Lichtenberg Ratio. When you cut this sheet of paper into two equal parts along its width, you get two sheets of paper where the ratio of length to width is 1 : 1 2 √2 . Explain why, and show that these sheets of paper also conform to the Lichtenberg Ratio. I
41 A stack of 25 cubes is shown below. Each cube has a volume of 8 cm 3
a Draw the full–scale front elevation and the fullscale right elevation of the stack of cubes. T2
b Explain whether 25 cubes can be stacked to form a larger cube. I
c Explain whether 343 cubes can be stacked to form a larger cube. I front
COmPuTaTiONal sKills
C2 A motorcycle racer completed 20 rounds on a 3.8 km circuit in 24 minutes.
i What was the motorcycle racer’s average speed in km/h?
ii A race car driver completed 15 rounds on the same circuit with an average speed of 210 km/h. How long did it take them to complete those 15 rounds? Give your answer in minutes and seconds.
Paper sizes
Worldwide, there are many different standards for paper sizes. In the Netherlands, the A series is generally used. The A4 size paper is part of the A series.
The largest A series size is A0. An A0 sheet of paper has dimensions of 1189 mm by 841 mm. The area of such a sheet of paper is exactly 1 m 2 . Each subsequent size in the A series is made by folding the larger sheet in half with the crease parallel to the shortest sides (see the figure on the right). If necessary, the length in mm is rounded down. For example, an A1 sheet of paper is 841 mm by 594 mm. The area of this sheet of paper is 0.5 m 2. The A11 size is the smallest sheet of paper in use. Some stamps are this size.
A series sheets of paper always have a length to width ratio of √2 : 1. This ratio is called the Lichtenberg Ratio. As the ratio of length to width is always the same, images can be enlarged or reduced to a different size in the A series without parts being cut off; nor is it necessary to stretch or compress the image.
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square root
second root taking the square root extracting the square root like radical terms
nth root
root of degree n third root
cube root
Lichtenberg Ratio
The A series paper sizes
Have you reached your goal?
R I know what like radical terms are and I know what n th roots are.
T1 I can add, subtract, multiply and divide square roots. I can also calculate n th roots.
T2 I can perform calculations with n th roots.
I I can show that the dimensions of paper sizes in the A series conform to the Lichtenberg Ratio.
1.3 Powers
GOAL You will learn how to perform calculations with powers of variables.
Powers 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 3, which is also called ‘the cube of two ’ or ‘two cubed’ and can be written as 23. The power 23 consists of base 2 and exponent 3. Variables can also be raised to a power. For example, a a a a = a 4 If you substitute a = –2 into the expression a 4, you get (−2) 4 = 16. A combination of variables can be raised to a power as well. For example, (ab)3 = ab ab ab = a a a b b b = a 3b 3
And (a b)2 = a b a b =
Examples
1
2
(ab) p = ap b
(a b)n = an bn
adding and subtracting powers Powers with the same base and the same exponent are called like terms. Like terms can be added or subtracted. For example, a 4 + a 4 = 2a 4, while a
+ a 4 and a 3 + b 3 cannot be simplified further.
Terms that consist of two or more variables are like terms only when all corresponding variables have the same exponent. For example, 2ab4 + 6ab4 = 8ab4, while 3a2b4 + 4 ab4 cannot be simplified further, because variable a has a different exponent in each of the terms.
Examples
EXERCISES — PRACTICE
Powers
44 Calculate without using a calculator. T1
a 24 d 33 − (− 3)3
b (−2)4 e (− 5)2 − 52
c 24 f 72 + (− 2)5
45 Write as a single power. T1
a a a a a a
b 2w · 2w · 2w
c ab · ab · ab · ab · ab · ab
d 5 x · 5 x · 5 x · 5 x · 5 x · 5 x · 5 x
46 Simplify. T1
a (3a)4 c (2ab)5
b (2 a)3 d (a b)2
47 Substitute x = −2 into the following expressions and calculate the result. T1
a 2x 3 c x 2 − x 3
b 1 4x 4 d (− 6x)2
48 Substitute a = 3 and b = −2 into the following expressions and calculate the result. T2
a a 2 − 6b c a 2b 4 − 2a 3b
b a 4 + b 3 + ab 2 d (2a 2b 3)2
adding and subtracting powers
49 Simplify. T1 a 4p 3 + 6p 3 − p 3 b 4y 7 − 9y 7 + y 7
c b 4 + a 2 − 5b 4 + 4
d 4k 2 − 19k 7 + 21k 7 e 3a 4 + 4a 3 − 5a 4 f 6a 5 − a 5 2 − b 2
50 Simplify. T1 a ab 5 − 2a 5b + ab 5
b 3p 2q 3 − 4pq 3 + 3 + 3pq 3
c 10x 4y 2 − 8 − 9x 4y 2 − x 4
d 8a 3c 4 − 2a 2c + 4a 3b 4 + 2a 2c
51 Simplify. T2 a x 3 4 + x 3 2 + x 3 6
b 0.5a 2b 2 − 3.5ab 2 + (ab)2
c (3p)2 − 10 − 6p 2 + 9p
d 5ab 2c − 10abc − 6ab 2c
l aws of exponents The following rules apply to calculations with powers:
When you multiply two powers that have the same base, you can add the exponents.
For example, a
When you divide two powers that have the same base, you can subtract the exponents.
For example, a 5 a 3 =
= a 2. This rule only applies when a ≠ 0. In this chapter you may assume that a ≠ 0.
When a power is raised to a power, you can multiply the exponents. For example, (a 3)
If the first power is a product of factors, you can raise each factor to the specified power and multiply the results.
For example, (ab 4)3
Examples
EXERCISES — PRACTICE
laws of exponents
52 Copy and fill in the correct operations.
Use +, , × or : R a a n a m = a n m
b a n a m = a n m
c (a p)q = a p q
53 Simplify. T1
a a 4 a 3 d 7x 3 2x 4
b x · x 2 e 2 x 3 · − 5x
c p 3 · p 5 f 9b 8 · − 3b 2
54 Simplify. T1
a a 6 a 2 d 36b 5 9b
b a 3 a 5 e 3x 9x 3
c x 10 x 3 f 60a 6 12a 3
55 Simplify. T2
a 3a 3 5a − 2a 2 7a 2
b 5a 4 8a 5 + 7a 2 2a 7
c 4x 3 9x 4 − 5x 5 7x
d 4x 3 + 9x 4 5x 5 − 7x 3
56 Simplify. T1
a (x 5)5 d (2x 3)6
b (x 2)4 e (x 4y 2)3
c (x 4)7 f (− 3x 2)4
57 Simplify. T2 a (2x 3y 2)4 d (7a 3)2 − 7a 6 b (− 4x 4y)3 e (− x)10 − (x 5)2
c (1 2x 3)4 f (2x 3)4 − (3x 4)3
58 The cuboid shown below has a volume of 12r 3 cm 3. It has a height of 3r cm and its base is a square. T2
a Calculate the length and width of the cuboid when r = 2.
b Set up a formula to calculate the length and width of the cuboid. Use r as the independent variable. 3r
Exploring
59 Write as a single power. T2
60 The net of a box with a square base is shown below. The box is twice as wide as it is high. T2
a Calculate the surface area of the box in dm 2 for a = 20 cm. Also calculate the volume of the box in dm 3 for the same value of a.
b Set up a formula to calculate the surface area of the box.
c Set up a formula to calculate the volume of the box.
investigating
62 Read the text Astronomical units on the next page. T2
a The average distance from Neptune to the Sun is about 4.5 109 km. Express this distance in astronomical units.
b The average distance from the dwarf planet Pluto to the Sun is 39.3 AU. How many kilometres is this? Write your answer in scientific notation and round to the nearest tenth.
63 Proxima Centauri is the star closest to our solar system. The distance from Earth to this star is about 267 500 AU. Distances to stars are measured in light years, because stars are so far away. A light year is the distance that light travels in a single year. The speed of light is about 300 000 km/s. Calculate how many light years Proxima Centauri lies away from Earth. Round your answer to the nearest tenth. I CO m P u TaT i ON al s K ills
C3 What is the total amount owed for the products below?
61 Copy and fill in the correct operations. Use +, , × or : I a 2a 3 3a 2 4a 5 = 10a 5
b 5a 3 a 3a 2 2a 3 = 6a 3
c 5a 4 a 2 a 3 = 5a 3 d 4p 6 6p 4 p 2 3p 8 20p 3 p 7 = p 10
Our solar system. The distances between the planets are drawn to scale. The inner orbits are enlarged.
a stronomical units
Distances within our solar system are often expressed in astronomical units (AU). One astronomical unit corresponds to the average distance from Earth to the Sun, which is about 150 million kilometres. In 2012, one astronomical unit was defined as exactly 149 597 870 700 metres. It is easier to compare distances in our solar system by expressing them in AU. The average distance from Uranus to the Sun is approximately equal to 2.9 billion km, which is equal to 2.9 109 1.5 · 108 ≈ 19.3 AU. This number shows that, on average, Uranus is about 19 times further away from the Sun than Earth.
Have you reached your goal?
R I know when I am allowed to add and subtract powers. I also know the laws of exponents.
T1 I can add, subtract, multiply and divide powers. I can also calculate a power of a power.
T2 I can simplify complex expressions with powers of variables.
I I can complete a calculation with powers by filling in the correct operations.
raising to a power the cube of base exponent like terms laws of exponents astronomical units
v OC abula RY
Sun Mars Earth
Mercury
1.4
Expanding brackets
GOAL You will learn how to expand brackets.
Two factors You can simplify an expression with two factors and brackets as follows. Use the FOIL–method when you have to multiply two binomials.
( b + c) = ab + ac (a + b)(c + d ) = ac + ad + bc + bd
Examples
1 −2(3x + 5) =
(2 x + 4)2 = (2 x + 4)(2 x + 4) = 4x2 + 8x + 8x + 16 = 4x2 + 16x + 16
Three factors When an expression consists of three factors, such as 3(2a − b)(4 a + 5b), you need to expand the brackets in steps. First, you multiply two of the factors. Then you multiply the result with the third factor. You can choose which two factors to start with. 3(2a − b)(4 a + 5b) = 3(8a 2 + 6ab − 5b 2) = 24a 2 + 18ab − 15b 2 or 3(2a − b)(4 a + 5b) = (6a − 3b)(4 a + 5b) = 24a 2 + 18ab − 15b 2
Examples
1 3(x − 4)(3x + 5) = 3(3x2 − 7x − 20) =
9x2 − 21x − 60 2 −(2b − 1)(10 + 5a) = −(20b + 10ab − 10 − 5a) = −20b − 10ab + 10 + 5a = −10ab + 5a − 20b + 10
3 6(2a − b)(a + b) = (12a − 6b)(a + b) = 12a 2 + 12ab − 6ab − 6b 2 = 12a 2 + 6ab − 6b 2 F
EXERCISES — PRACTICE
Two factors
64 Simplify.
a 6(z + 3) T1 d 6y(−4y 2 + 5) T2
b (−3 − z) T1 e (8 − x) 7x T2
c 2 x(5 − x) T1 f p4(p2 + p) T2
65 Simplify. T1
a (x + 5)(x + 4)
b (x + 8)(x − 9)
c (2 x + 1)(3x − 1)
d (− 3x + 1)2
66 Simplify. T2
a 3 − (9x − 3)
b 4p(3q − 5) − 7q(2p − 1)
c a(a − 7) − (a − 7)(a + 3)
d 20 − (x + 4)2
67 Set up a formula to calculate the area A of the orange part of the rectangle shown below. Expand the brackets. T1
Three factors
68 Simplify. T1
a 2(x + 6)(4x − 2)
b 4( p − 3)(5p − 3)
c (3p + 7)(2 − 9q)
d 3(a − b)2
69 A container measuring 2a dm by a + 3 dm by a – 1 dm is shown below. T2
a Set up a formula to calculate the volume V of the container. Expand the brackets.
b Why must a be greater than 1?
c Calculate the volume of the container for a = 3.
s pecial binomial products You should memorise the following three special binomial products. They can be used to quickly expand brackets or factorise expressions.
First binomial product :
Second binomial product :
Third binomial product
The third binomial product is referred to as the ‘difference of two squares’.
Note: when simplifying expressions of the form 3(a + 1)2, you first need to expand the brackets and then multiply the result with the factor in front of the brackets: 3(
Examples
EXERCISES — PRACTICE
special binomial products
70 Combine each of the expressions on the left with the correct binomial product on the right. R
A a 2 − 2ab + b 2 1 (a − b)(a + b)
b a 2 − b 2 2 (a + b)2
C a 2 + 2ab + b 2 3 (a − b)2
71 Simplify. T1
a ( p + 1)2 d (x − 2)(x + 2)
b (2a − b)2 e (3a − 2b)2
c (7k − 5)2 f (8x − 2y)(8x + 2y)
72 a Use the figure below to show that (a + b)2 = a2 + 2ab + b2 . T1
a a b b
b Make a similar drawing for each of the other two special binomial products. I
73 Simplify. T2
a (x 2 + 1)2 d (p 3 + 1)2 b (6a 2 − 2b 2)2 e (3x 3 − 1)(3x 3 + 1)
c (x2 − 10)( x2 + 10) f (z 6 − z)2
74 Copy and fill in the blanks. T2
a a 2 + … a + 36 = (a + …)2 b … − 9 = (p − …)(p + …)
c 1 4 q 2 − … q + … = (… − 7)2
75 The first binomial product (a + b)2 = a 2 + 2ab + b 2 can be used to quickly calculate squares: 312 = (30 + 1)(30 + 1) = 302 + 2 · 30 · 1 + 12 = 900 + 60 + 1 = 961
Calculate the following squares in a similar way. T2 a 232 c 992 b 542 d 1042
Exploring
76 Simplify. T2
a 3 − 6 ( p − 1)2 + 8p 2
b x(x 2 + 2)(x 2 − 3) − 5(x 4 + 4)
c 2p(p 2 − 4)(p 2 + 5) − 2(p 3 − 1)
d ( √a + √b )( √a − √b )
e (x − 1)(x + 2)(x − 3)
77 Two sides of a square measuring 4 cm by 4 cm are repeatedly expanded by a 1 cm wide strip to create consecutively larger squares. T2 4 1 1 1 4 1 1 4 1 4 1
a Copy the table and fill in the missing values.
b Set up a formula for the area A of square n
c Set up a formula for the area of the last strip added to square n. Simplify this formula as much as possible.
78 Sometimes the difference of two squares (a – b)(a + b) = a2 − b2 can be used to quickly calculate the product of two numbers: 98 102 = (100 − 2)(100 + 2) = 1002 − 22 = 10 000 − 4 = 9996
Calculate the following products in an efficient way. I a 17 23 d 97 103 b 28 32 e 812 − 802 c 54 46 f 0.91 1.09
CO m P u TaT i ON al s K ills
C4 Calculate in an efficient way.
i 4 · 37 · 25 iv 99 · 99 ii 4 · 67 + 6 · 67 v 23 · 38 − 3 · 38 iii 98 · 45 vi 17 · 102
investigating
79 Several square tiling patterns consisting of grey and orange tiles are shown above.
a Copy the table and fill in the missing values. T2
tiling pattern n 1 2 3 4 5
number of orange tiles
b Set up a formula for the number of orange tiles o in tiling pattern n T2
c Set up a formula for the number of grey tiles g. Simplify as much as possible. I
d Determine if any of the tiling patterns in this sequence has an equal number of grey and orange tiles. I
e Calculate 1 + 2 + 3 + 4 + + 100. I
FOIL–method first binomial product second binomial product third binomial product difference of two squares 1 2 3 4
Hint : look at the number of grey tiles in one of the corners of the tiling patterns shown at the top of this page.
Have you reached your goal?
R I know what the FOIL–method is and I know the three special binomial products.
T1 I can expand brackets.
T2 I can simplify complex expressions that contain brackets.
I I can use the special binomial products to efficiently calculate the product of two numbers.
vOCabulaRY
Fractions with variables
GOAL You will learn how to perform calculations with fractions that contain variables.
simplifying Fractions can contain variables. Fractions with variables need to be simplified as much as possible. For example, 4a 2b a = 4 a a b a = 4 ab The denominator of a fraction cannot be 0, because division by zero is undefined. Therefore, a in the expression 4a 2b a cannot be equal to 0 and this expression can only be simplified when a is not equal to 0. In this chapter you may assume that the variable by which you divide is never equal to 0.
Examples
1
adding and subtracting Fractions with the same denominator are called like fractions. Like fractions can be added or subtracted. When like fractions are added or subtracted, their numerators are added or subtracted. Their denominators stay the same. For example, 6 a + 2 a =
Unlike fractions have to be converted to like fractions before they can be added or subtracted. This can be done by multiplying both the numerator and the denominator by the same number or the same variable. For example, 1
Examples
1 6 a + 11 a = 6 + 11 a = 17 a 2
EXERCISES — PRACTICE
simplifying
80 Simplify the following fractions. T1
a 5ab 25b c ab 3 ab
b 33a 2 11ab d 12abc 6b
81 Simplify the following fractions.
a a 5b 3 a 2b T1 c 8a 4b 7 2a 3b 3 T1
b a 2b 3 a 5b 2 T1 d (ab)5 3ab 2 T2
82 Calculate the length of the rectangle shown below. T2 12a2b ? 3a
adding and subtracting
83 Calculate. T1
a 3 11 + 7 11 d 1 2 + 5 9
b 12 19 − 17 19 e 5 8 − 2 3
c 1 5 7 + 2 3 7 f 1 2 3 + 5 6
84 Write as a single fraction. T1
a 15 a + 3 a c b + 5 a − 1 a
b 9b a − 2 a d b a + 3b a
85 Copy and fill in the blanks. T1 a 3 4x + 2 x = 3 4x + … 4x = … b 2 6xy − 5 2y = 2 6xy − … 6xy = … c 7 x + 9 2x 2 = … 2x 2 + … 2x 2 = d 8 3p − 2 = 8 3p − 2 1 = 8 3p + … 3p = …
86 Write as a single fraction. T1
a 9 a + 2 ab e 2 x − 3 y
b a 18b + 6 6b f 6 3x − x 2y
c 7 x + 9 x 2 g 8 x + 2 3 d 8 2b − 4 5b h 1 − 4 a 2
87 Write as a single fraction. T2
m ultiplying and dividing fractions Fractions with variables are multiplied and divided in the same way as fractions without variables.
Multiplying To multiply two fractions, multiply their numerators and their denominators. For example,
Dividing To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.
Examples of multiplications with fractions
1 5 a · 2 9 = 5 · 2 a 9 = 10 9a
2
Examples of divisions with fractions 1 12p q : q 3 = 12p q · 3 q = 12p · 3 q · q = 36p q 2
2
a b c d = ac bd a b c d = a b : c d = a b d c = ad bc
EXERCISES — PRACTICE
multiplying and dividing fractions
88 Calculate. T1
a 4 9 3 8 c 1 2 5 2 7
b 2 3 3 4 d 4 11 1 1 10
89 Write as a single fraction and simplify as much as possible. T1
a 2a 9 2 7 d 2p 9q 3q 5
b 2a 5b · 3a 7 e 3 3 7 · 7q 8p
c 3a 7c · 2b 4 f 6x y · 5x
90 (a b)2 = a b · a b = a 2 b 2. Calculate using a similar method. a (2 x 3y)2 T1 c (3x 3 y 2 )3 T2 b ( a 2 b 3)2 T1 d ( a 2 2b)3 T2
91 Calculate. T1
a 6 7 : 3 5 d 4 1 2 : 1 1 5
b 3 16 : 1 2 e 3 3 4 : 1 1 2
c 15 : 3 7 f 4 15 : 5
92 Write as a single fraction and simplify as much as possible. T1 a 4 a : 5 12 d 5a 2b : b 2a b 3a 4b : b 3 e 10 3 a c 3a 4b : 2a f 3a 4 a
93 Write as a single fraction and simplify as much as possible. T2 a 3x 4y · x 3y 2 d 5a 3 2b 2 : b 6 2a 7 b 2x 5 5y · 2y 2 9x 2 e p 4 q : p q 2
c 6a b 3 5a 3 f x 4 : 2x 2 y 4
Exploring
94 Copy and fill in the blanks. T2
a + 5 a = 16 a d 6a 2 b = 2a 5 5b 4
b 7 a − = 3 2a e 4 a b 2 = b a 5
c (…)2 = 4a 4 25
95 A rectangle with an area of 10a2b2 is given. Copy the table and fill in the missing values. T2 width 10a 2 5b 5b 3 length a 2 ab 2 2a 2b
96 When you simplify 5x x , you get 5, provided that x ≠ 0.
a Explain why the condition x ≠ 0 needs to be included in the simplification. T1
b Simplify the following expressions. Also write down any necessary conditions.
I 2 x · 5 2 x T2
II 5(x + 3) x + 3 T2
III 7x − 7 x − 1 I
IV x 2 − 4 x − 2 I
investigating
97 Read the text Pythagorean tuning on the next page. T2
a Suppose you use a 10 cm string for musical note A. What are the lengths of the strings for the other notes?
b The text explains how to get a D by going up a perfect fourth from an A. Note G can be found by going down a perfect fifth from a D. Explain how, starting with an A, you can get all the other notes.
c What is the ratio of the lengths of the strings of notes E and D?
d Calculate the ratio of the lengths of the strings for each pair of consequent notes. What do you notice?
98 A xylophone is not a string instrument, but a percussion instrument. However, you can use Pythagorean tuning on a xylophone as well. The areas of the keys need to have the same ratios as the lengths of the strings of a string instrument. Draw the top view of a xylophone in Pythagorean tuning. I
CO m P u TaT i ON al s K ills
C5 A unit fraction is a fraction where the numerator is 1 and the denominator is a positive integer. For example, 1 5 and 1 7 are unit fractions.
i Write 3 4 as the sum of different unit fractions.
ii Write 1 2 as the sum of different unit fractions.
iii Give the largest possible unit fraction that is smaller than 4 27
Pythagorean tuning
The Greek philosopher Pythagoras, who lived in the sixth century BC, is known worldwide for the theorem named after him. Pythagoras also contributed to music by creating the so called Pythagorean tuning. Tuning in music is a process to establish the pitch of tones by using specific ratios between tones.
For his research on the connection between pitch and the strings’ lengths, Pythagoras used a monochord, a musical instrument with one or more strings. He discovered that at specific ratios between the lengths of the strings, the tones sounded pleasant together. He used these ratios to create the notes A to G. The ratios that he used led to the current western interval system with the octave, perfect fifth and perfect fourth. In the table on the right you can see by which factor the length of a string is multiplied to get a tone that goes up an octave, a perfect fifth or a perfect fourth. Pythagoras used note A as a base tone. The D, a fourth up, is obtained by shortening the Astring by a fourth. If the length of the Astring is 1, then the length of the D string is equal to 1 · 3 4 = 3
v OC abula RY like fractions unlike fractions reciprocal tuning
A G is a fifth lower than a D. Pythagoras could get a G by increasing the length of the D string by one and a half. Therefore, if the length of the Astring is 1, the length of a G string is equal to 3 4 3 2 = 9 8. In this way, Pythagoras found the different notes by establishing the ratios of the strings’ lengths. The stave at the top of this page shows the notes and the corresponding lengths of the strings, based on Pythagorean tuning.
string length what happens to the tone
× 2 octave down
× 1 2 octave up × 3 2 perfect fifth down
× 2 3 perfect fifth up
× 4 3 perfect fourth down × 3 4 perfect fourth up
Have you reached your goal?
R I know that the same rules apply to fractions with variables and fractions without variables.
T1 I can perform calculations with fractions that contain variables.
T2 I can determine which fraction is needed to complete a calculation.
I I can simplify complex fractions with variables.
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
99 A bicycle has a gearwheel with 48 teeth on the bottom bracket axle. A gearwheel on the rear axle has 16 teeth.
a How many times does the back wheel rotate when the pedals of this bicycle rotate 100 times? T1
b The diameter of the back wheel is 27.5 inches. Calculate the bicycle’s gear inches. Round your answer to the nearest inch. T1
c Calculate your speed in km/h when you rotate the pedals 40 times per minute. Round your answer to the nearest tenth. 1 inch = 2.54 cm. T2
§ 1.1
100 a What is the smallest prime number? R
b Give all prime numbers between 10 and 20. T1
c Give the prime factorisation of 160. T1
101 For each of the numbers below, explain whether it is a natural number, an integer, a rational number or an irrational number. Note: each number can have multiple correct answers. T1
§ 1.2
103 Calculate. Do not use a calculator. T1
2 + 3 √5 − 6 + √5
3 √7 + 5 √2 − 2 √2 − √7
2 √5 8 √7
104 Simplify. Do not use a calculator. T1
105 Calculate if possible. If not possible, explain why. Do not use a calculator. T1
102 Consider the fraction 525 3000 T2
a Give the prime factorisation of both the numerator and the denominator.
b Simplify the fraction 525 3000 as much as possible.
106 A figure consists of three squares with areas of 25, 17 and 3. What is the perimeter of this figure? Simplify your answer as much as possible. I 25 17 3
§ 1.3
107 Simplify. T1
108 Write as a single power. T1
a x 2 x 5 c (x 4)5
b x 8 x 3
d (x 2y 3)3 § 1.4
109 Simplify. T1
a 4(2a − 3b) d (4 a − 5b)( − 2a + b)
b 2 x(6x − x 2) e 3(x − y)(3x − 2y)
c p4( p2 + 2p 3) f 5a (a − b)2
110 A rectangle is 5 cm longer than it is wide. Show that the rectangle has an area of x2 – 5x cm 2, where x is the length of the rectangle. T2
§ 1.5
111 Write as a single fraction. T1
a 7 2 x + 2 x c 3 − 6 p 3
b 8 x − 4 x 2 d 2 a + 3 5b
112 Write as a single fraction and simplify.
a 7 2a 4b a T1 c 3p 4 q : 9p 2 T1
b 6x 3 5y 4 2 x 9y 2 T2 d 8p 2 q : 2q 3 T2
Chapter 1
113 Calculate using your calculator. Round your answer to the nearest hundredth. T2
a (− 2 √3 )5
b 2 3√ 12 − 5( 4√3 − 1)
c 1 5 12 − 3 3 7
114 Simplify. T2
a 6p 2( p − 1)2 − 3p 2
b (x 2 − x 3)(x 2 − x 4)
c 1 − q(q − 1) + 2(q − 1)2
d ( √a + 2)( √a − 2)
e (1 2 a + 2) (a − 1)2
115 A rectangle with an area of 18x + 54 is shown below. The blanks represent integers. Give all of the options. I
x + ... 18x + 54
116 Suppose you can only jump 11 2 units to the right or 2 3 units to the left on a number line. You start at 0.
a You jump twice to the right and once to the left. At which number do you end up? T2
b Explain which jumps are needed to end up at –1 when you start at 0. T2
c Explain whether you can end up at √2 if you start at 0. I