Focus Smart Plus Maths Textbook M3 samplebook

BCB033038

Focus Smart Plus Mathematics Textbook covers the entire range of topics included in the Basic Education Curriculum B.E. 2551 (Revised Edition B.E. 2560). Notes and plenty of exercises are given to help students understand and apply the concepts in daily life.

BCB033038 978-616-541-314-5

9 786165 413145

Based on the Basic Education Curriculum B.E. 2551 (Revised Edition B.E. 2560)

© Pelangi Publishing (Thailand) Co., Ltd. 2019 All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means without permission of Pelangi Publishing (Thailand) Co., Ltd. 2019

BCB033038_FSPMathematics2018TBM3_openpage.indd 1

ISBN 978-616-541-314-5 First Published 2562

22/3/2562 BE 11:23

Contents Chapter

Chapter

Chapter

1

2 3

Linear Inequalities 1.1

Inequality

1

1.2

Linear Inequalities in One Variable

4

1.3

Performing Computation on Inequalities

8

1.4

Solving Inequalities in One Variable

11

1.5

Systems of Linear Inequalities in One Variable

17

Mastery Practice

24

Quadratic Equations in One Variable 2.1

Quadratic Equations

25

2.2

Roots of Quadratic Equations in One Variable

29

Chapter

4

5

26

Mastery Practice

37

Systems of Linear Equations

38

3.1

Systems of Linear Equations in Two Variables

3.2

Solving Problems Involving Systems of Linear

Chapter

2

Equation in Two Variables

39 44

Mastery Practice

48

Quadratic Functions 4.1

Quadratic Functions and Their Graphs

49

4.2

Maximum and Minimum Values of Quadratic Functions

56

4.3

Sketching Graphs of Quadratic Functions

58

50

Mastery Practice

62

Factorization of Polynomials 5.1

Factorizing using the Special Product Method

64

5.2

Factorizing using the Highest Common Factor (HCF)

69

5.3

Factorizing using the Grouping Method

72

5.4

Factorizing using the Synthetic Division Method

73

Mastery Practice

65

79

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

6 7 8

9

10 11

Surface Areas and Volumes of Pyramids, Cones and Spheres 6.1

Surface Areas of Pyramids, Cones and Spheres

80

6.2

Volumes of Pyramids, Cones and Spheres

86

81

Mastery Practice

94

Similarity 7.1

Scale Drawings

95

7.2

Similar Triangles

106

96

Mastery Practice

112

Trigonometric Ratios 8.1

Tangent

114

8.2

Sine

119

8.3

Cosine

122

8.4

Values of Tangent, Sine and Cosine

125

115

Mastery Practice

134

Circles 9.1

Parts of a Circle

136

9.2

Angles in a Circle

139

9.3

Cyclic Quadrilaterals

146

9.4

Angles between Tangents and Cords

150

137

Mastery Practice

158

Box Plots

160

10.2 Interpreting Box Plots

166

Mastery Practice

169

Probability

171

11.2 Probability

175

11.3 Outcomes from Independent Events

182

Mastery Practice

188

10.1 Constructing Box Plots

11.1 Events and Outcomes

161

172

Special Features

in This Book

Learning Outcomes States the learning objectives of each chapter.

Flashback

Test Yoursel f Evaluates the understanding of the students for every subtopic.

Lists the important mathematical terminologies of each chapter.

Helps students to recall the basic concepts for the chapter.

Math Online Provides direct access to useful websites by scanning the QR codes given.

Consists of brief and concise notes that summarise the concepts learnt in each chapter.

Mastery Practice Provides subjective questions covering the entire learning outcomes of each chapter.

Points out the important tips for students to take note.

Example

Points out the common mistakes that students make and the correct ways of answering questions.

Consists of sample questions with complete and comprehensive solutions.

Provides direct access to the interactive exercises by scanning the QR codes given.

1 Chapter

Linear Inequalities

Flashback 1. Find the values of y.

(a) y = 3x + 5; x = 2

(b) y =

x

5

+ 2; x = 10

(c) y = 7x – 12; x = 3

By the end of this chapter, you should be able to • apply knowledge of linear inequalities with one variable for problem solving, as well as aware of the validity of the answer.

(d) 3y + 4x = 5; x = 7 (e) (f)

6 y

x = 2; x = 9 3x – y 1 x+y = 3 ;x=1

2. Express x in terms of y. (a) 4x + 3y = 6

(b) 16x – 8y = 10

(c) 13x = 16x + 5y + 2

(d) 4y + 3x = 6y + 2x + 5

y

(e) y = 9 6 – 4x 2. (a) y = 3 –(3x + 2) (c) y = 5 –2(x + 2) 3 (c) y = 9

Math Online Visit these websites to know more about this chapter.

1. (a) y = 11 Answers:

(f)

2x – 3y =4 x+1 4(x + 2y) = 12

(e) y =

(e)

Mathematics Focus Smart + MATHAYOM

Chapter 1 Linear Inequalities

3

1

(f) y = x (d) y =

2

x–5

(b) y = 4 2 (d) y = –7 3 (f) y = 2 8x – 5 4

(b) y =

1.1 Inequality A

Identifying the relationship ‘greater than’ (>) and ‘less than’ (<)

Inequality is the relationship between two ­unequal quantities. The symbols used are: (a) > read as greater than (b) < read as less than

Example 1 Identify the relationship ‘greater than’ or ‘less than’ based on the following situations. (a) The temperature of the water is below 40°C. (b) Suda’s sales for today is more than 10,000 Baht.

Solution (a) Less than

Below means ‘less than’.

(b) Greater than

More than means ‘greater than’.

Example 2 Write the relationship between the two numbers in the following sets of numbers using the symbol ‘>’ or ‘<’. (a) 3.142,

22 7

5 2 (b) 1 , 1 7 3

Solution 22 = 3.1428 …

7 22 22 Therefore, 3.142 < or > 3.142 7 7

(a)

(b) 1

5 15 =1 7 21

2 14 =1 3 21 5 5 2 2 Therefore, 1 > 1 or 1 < 1 7 3 3 7 and 1

Try Questions 1 & 2 in Test Yourself 1.1

Mathematics

2

Focus Smart + MATHAYOM

3

B

Identifying relationship ‘greater than or equal to’ (>) and ‘less than or equal to’ (<)

The symbol for: (a) greater than or equal to is ‘>’ (b) less than or equal to is ‘<’

Example 3 Identify the relationship ‘greater than or equal to’ or ‘less than or equal to’ based on each of the ­following situations. (a) The speed limit on the highway x is 80 km/h. (b) The minimum mass of a box of chocolate is 500 g.

Solution (a) The speed on the highway x can be less than or equal to 80 km/h. (b) The mass of a box of chocolate can be greater than or equal to 500 g.

Try Question 3 in Test Yourself 1.1

Test Yourself

1.1

1. Identify the relationship ‘greater than’ or ‘less than’ based on each of the situations below. (a) The body length of that baby is shorter than 30 cm. (b) The school netball team scored under 12 goals. (c) The height cleared by that high jumper is above 2.1 m.

2. Write the relationship between each of the ­following pairs of numbers using the symbol ‘>’ or ‘<’. (a) 3.084, 3.48 (b) –7, –13 1 (c) 0.12, 8 4 7 , (d) 7 13 3. Identify the relationship ‘greater than or equal to’ or ‘less than or equal to’ based on each of the following situations. (a) A company made a maximum profit of 3.5 million Baht. (b) A senior citizen must be at least 55 years old. (c) The price of each unit of double storey house is 1,800,000 Baht and above.

Mathematics Focus Smart + MATHAYOM

Chapter 1 Linear Inequalities

3

3

Linear Inequalities in One Variable

1.2 A

Determining if a given relationship is a linear inequality in one variable

A linear inequality in one variable is an inequality with a variable or unknown raised to the power of one.

Example 4 Determine whether each of the following ­relationship is a linear inequality in one variable.

1 (a) x – 2 < 3 (b) y > 2x + 4 (c) >x+6

x

Solution (a) Yes (c) No

(b) No 1

x

>x+6

y > 2x + 4 Two variables

Not a linear term Try Question 1 in Test Yourself 1.2

B

Determining the possible solutions for a given linear inequality in one variable

Example 5 Determine the possible solutions for each of the following inequalities, if x is an integer.

1 (c) x > –3 (a) x > 42 (b) x < 2 2

Solution (a) If x is an integer, the possible solutions for x that are greater than 42 are 43, 44, 45, …

1 (b) If x is an integer, the possible solutions for x that are less than 2 2 are 2, 1, 0, –1, … (c) If x is an integer, the possible solutions for x that are greater than or equal to –3 are –3, –2, –1, 0, 1, …

Try Question 2 in Test Yourself 1.2

Mathematics

4

Focus Smart + MATHAYOM

3

C

Representing a linear inequality on a number line and vice versa

A linear inequality can be represented on a ­number line. For example, (i)

x > a

x>a

a ‘o’ means a is not a value for x.

(ii)

xb

x>b

b ‘ ’ means b is a value for x.

(iii)

x< c

c<x

c

(iv)

xd

xd

d

Example 6 Represent each of the following inequalities on a number line. (a) (c)

x>3

x < – 12

(b)

x > –2

(d)

x < 4.5

Solution (a)

x > 3 2

(c)

3

4

(b)

x > –2 –3 –2 –1

5

(d)

x < – –1 2

–3 –2 –1– –12 0

0

x < 4.5 2

3

4 4.5 5

Try Question 3 in Test Yourself 1.2

Mathematics Focus Smart + MATHAYOM

Chapter 1 Linear Inequalities

3

5

When given a number line, a linear inequality can be written by observing (a) the direction of the arrow, (b) the symbol ‘o’ and ‘ ’.

For examples, (i)

x>a

a Arrow towards the right (→) and the symbol ‘o’

(ii)

x>a

a Arrow towards the right (→) and the symbol ‘ ’

(iii)

x<a

a Arrow towards the left (←) and the symbol ‘o’

(iv)

xa

a Arrow towards the left (←) and the symbol ‘ ’

Example 7 Write an inequality for each of the following ­representations on the number lines. (b) (a) 9

–7 –6 –5 –4

10 11 12

(c)

(d) 0

1

2 2.3 3

–1 – –14 0

Solution (a) x >10

(b) x < –5

(c) x < 2.3

(d) x > –

Try Question 4 in Test Yourself 1.2

Mathematics

6

Focus Smart + MATHAYOM

3

1 4

1

2

D

Words maximum

Symbol

minimum

>

at least

>

<

at most

<

not more than

<

not less than

>

Constructing linear inequalities

Linear inequalities can be constructed from given information by following the steps below.

1 Identify the variable and represent it with a suitable letter.

2 Use the symbols ‘>’, ‘<’, ‘>’ or ‘<’ based on the given information.

3 Write the linear inequality.

Example 8 Construct a linear inequality for each of the ­following information. (a) The maximum height of a vehicle that can pass below this bridge is 3.5 m. (b) At least ten pupils in the school were offered a place in the boarding school. (c) Sunee has read more than 4 storybooks this week.

Solution (a) Let h m be the height of a vehicle that can pass below this bridge.

Therefore, h < 3.5 (b) Let n be the number of pupils that were offered a place in the boarding school.

Therefore, n > 10 (c) Let m be the number of storybooks read by Sunee this week.

Therefore, m > 4 Try Question 5 in Test Yourself 1.2

Test Yourself

1.2

1. Determine whether each of the following ­relationships is a linear inequality in one variable. (a) 2x – 1 > x + 3

(b) x 2  4

(c) p – 1  q

(d) xy > –2

2. Determine the possible solutions for each of the following linear inequalities if x is an integer.

1 (b) x < – 4 (a) x > 3.5 2 (d) x  8 (c) x  –5

Mathematics Focus Smart + MATHAYOM

Chapter 1 Linear Inequalities

3

7

3. Represent each of the following inequalities on a number line. (b) x > – 4 (a) x  5 1 (c) x < 1.5 (d) x  – 3 4 4. Write an inequality for each of the following. (b) (a) 11

–9 –8 –7 –6

(c)

12 13 14

(d) 3 3 –52 4

2

–8 –7 –6 –5.6–5

5

5. Construct a linear inequality for each of the ­following information. (a) The minimum mark scored by the pupils in the Mathematics test was 56. (b) Sakda’s collections for today is not less than 5,000 Baht. (c) Mali ate less than 100 calories each meal. (d) Not more than 20 teachers have seen the show.

Computation on 1.3 Performing Inequalities A

Adding and subtracting a number from both sides of an inequality

When a number is added to or subtracted from both sides of an inequality, the condition of the inequality is unchanged.

Example 9 State a new inequality when (a) 5 is added to both sides of the inequality 1 < 7, (b) 2 is subtracted from both sides of the ­inequality x < –3.

Solution (a)

1<7 1 +5 <7 +5 6 < 12 New inequality is 6 < 12.

Try Questions 1 & 2 in Test Yourself 1.3

Mathematics

8

Focus Smart + MATHAYOM

3

(b)

x < –3 x – 2 < –3 – 2 x – 2 < –5

New inequality is x – 2 < –5.

B

When we multiply or divide both sides of an inequality by a negative number, all the signs of symbols will change, that are,

When we multiply or divide both sides of an ­inequality by: (a) the same positive number, the condition of the inequality is unchanged. (b) the same negative number, the inequality is ­reversed.

‘+’ → ‘–’, ‘–’ → ‘+’, ‘>’ → ‘<’, ‘<’ → ‘>’ For example,

–3 < 1 [multiply by –2] –3 × (–2) > 1 × (–2) 6 > –2

Multiplying and dividing both sides of an inequality by a number

Example 10 State a new inequality for each of the following inequalities when both sides of the inequality are multiplied or divided by the number shown in the brackets. (a) – 4 < 1 [multiply by 2] (b) 3x > –12 [divide by – 3]

Solution (a)

–4 < 1 –4 × 2 < 1 × 2 –8 < 2

(b)

3x > –12 –12 3x < –3 –3 –x < 4

Inequality sign is unchanged.

Inequality is unchanged when multiplied by a positive number.

Inequality sign is changed.

Inequality is reversed when divided by a negative number.

Try Questions 3 & 4 in Test Yourself 1.3

C

Constructing inequalities from given information

To construct inequalities from given information, we follow the steps below.

1 Construct the basic inequality in the form of x > a, x < a, x > a or x < a where a is a number.

2 Based on the information given, ‘+’, ‘–’, ‘×’ or ‘÷’ both sides of the inequality by the same number to obtain a new inequality.

3 Simplify the new inequality.

Mathematics Focus Smart + MATHAYOM

Chapter 1 Linear Inequalities

3

9

Example 11 Few months ago, Niti bought two fish as pets. One of them has a mass of x kg and was heavier than the other fish that has a mass of 45 g. Now the mass of each fish is five times as heavy as compared to a few months ago.

Construct a relationship between the present masses of the two fish.

Solution 45

x > 1,000

45 1,000 Therefore, 5x > 0.225 5x > 5 ×

Change ‘g’ to ‘kg’.

Mass of the fish is 5 times the original mass.

Example 12 Mali bought x pens at a price of 45 Baht each. She paid 300 Baht and received a balance that is more than 30 Baht. Form an inequality for x.

C x>6 A x>6 D x<6 B x<6

Solution Payment for x pens < (300 – 30) Baht 45 x < 270 270 45 x < 45 45 x <6 Answer : D Try Questions 5 – 7 in Test Yourself 1.3

Test Yourself

1.3

1. State a new inequality for each of the following inequalities when the number in brackets is added to both sides of the inequality. (a) 2 < 3 [7] (b) 6 > – 4 [2] (c) x >12 [8] (d) y < –10.5 [5] 2. State a new inequality when the number in brackets is subtracted from both sides of each of the following inequalities.

22 (a) 16 > 11 [12] (b) 3.14 < [2] 7 1 1 (c) x > 7.6 [1.8] (d) y < –9 [ ] 2 3 Mathematics

10

Focus Smart + MATHAYOM

3

When constructing an inequality, ensure that the units used are the same.

3. State a new inequality for each of the following inequalities when both sides of the inequality are multiplied by the number shown in brackets. (b) x > –2 [5] (a) 2 < 7 [7] (c) 9 > –1 [–3] (d) 0 > –5 [–2] (e) –10 < –6 [ –

1 ] 2

(f)

y < –2.5 [–4]

4. State a new inequality for each of the following inequalities when both sides of the inequality are divided by the number shown in brackets. (a) 40 > 24 [8] (c) x < –63 [9]

(b) –72 < –54 [18] (d) –15 < 5 [–5]

(e) –27 > –72 [–9]

(f)

y > 1 [–

1 ] 2

5. Dej and Ake had x litres and 5 litres of petrol in the petrol tanks of their cars respectively. The amount of petrol in Dej’s tank is at least the amount of Ake’s. Construct an inequality ­showing the ­relationship between the amount of petrol in their tanks if each of them had filled 20 litres of petrol into their tanks.

6. Nuch and Kanda has y Baht and 90 Baht in their purses respectively.

Nuch has less money than Kanda. Each of them donates 50 Baht to a charity. Construct a relationship between the amount of money left in Nuch’s and Kanda’s purses.

7. Wandee bought 30 metres of ribbon and l metres of cloth. The length of the cloth is not longer than that of the ribbon. She divided the ribbon and cloth equally to decorate the four noticeboards in her class. Construct a ­relationship between the length of the ribbon and cloth for each noticeboard.

Inequalities in One 1.4 Solving Variable A

Solving a linear inequality in one variable by adding or subtracting a number

When given an inequality in the form of: (a) x + k > m where x is a variable, m and k are constants, we can solve it by subtracting k from both sides of the inequality. (b) x – k > m, we can solve it by adding k to both sides of the ­inequality.

We can verify or check the results by selecting a value for the variable that satisfies the new inequality and substitute it into the given inequality.

Mathematics Focus Smart + MATHAYOM

Chapter 1 Linear Inequalities

3

11

BCB033038

Focus Smart Plus Mathematics Textbook covers the entire range of topics included in the Basic Education Curriculum B.E. 2551 (Revised Edition B.E. 2560). Notes and plenty of exercises are given to help students understand and apply the concepts in daily life.

BCB033038 978-616-541-314-5

9 786165 413145

Based on the Basic Education Curriculum B.E. 2551 (Revised Edition B.E. 2560)