Collins Int. LS SB 8 - Look Inside

13.5mm spine

Collins

STAGE 8: STUDENT’S BOOK

Series editor: Alastair Duncombe Authors: Belle Cottingham, Alastair Duncombe, Rob Ellis, Amanda George, Brian Speed This Student’s Book provides comprehensive coverage of Stage 8 of the Cambridge Lower Secondary Mathematics curriculum framework (0862) from 2020 through 25 clear and supportive chapters.

The course encourages students to make links between the topics, build on what they already know, and develop their confidence in Thinking and Working Mathematically.

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Cambridge Lower Secondary Maths STAGE 8: STUDENT’S BOOK

Cambridge Lower Secondary Maths

Cambridge Lower Secondary

Maths

STAGE 8: STUDENT’S BOOK

ISBN 978-0-00-837854-7

9 780008 378547

04/01/2024 10:50

2.0 Getting started Take any five points on the circumference of a circle and join them to make a five-sided polygon.

Two diagonals can be drawn from any vertex of the polygon.

•

How many diagonals can be drawn from one vertex in a seven-sided polygon?

•

Find out how many diagonals could be drawn from one vertex in a polygon with 12 sides, without the need to draw it.

•

How many diagonals altogether (from all vertices) can be drawn in a seven-sided polygon?

•

How many diagonals altogether can be drawn in a 15-sided polygon, without the need to draw it?

2.1 Quadrilaterals Key terms A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal.

The diagonals of a quadrilateral bisect each other if one diagonal cuts the other exactly in half.

diagonals

Properties of quadrilaterals A square has four equal sides and four right angles. Opposite sides are parallel.

A rhombus has four equal sides. Opposite sides are parallel and opposite angles are equal.

The diagonals of a square are equal in length, are perpendicular and bisect each other.

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The diagonals of a rhombus are perpendicular and bisect each other.

A rectangle has four right angles.

Opposite sides are parallel and equal in length.

In a parallelogram, opposite sides are parallel and equal in length. Opposite angles are equal.

The diagonals of a rectangle are equal in length and bisect each other.

The diagonals of a parallelogram bisect each other.

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10 A bicycle wheel has diameter 70 cm. a) Find how far the bicycle travels if the wheel makes 20 revolutions. b) Find the number of complete revolutions the wheel makes when the bicycle travels 1 km.

11 A cycle track is circular. The inner radius is 60 m and the track is 5 m wide.

Not to Scale

Kat cycles on the outside edge of the track.

Eve cycles on the inside edge of the track.

Find how much further Kat cycles than Eve in making one circuit of the track.

5m

60 m

Consolidation exercise

1 Write down the name of the quadrilateral that is being described in each part. There may be more than one answer for each. a) Opposite sides are equal in length. The diagonals bisect each other and are equal in length. b) Opposite sides are parallel. Every side is the same length. c) The diagonals intersect at right angles with the shorter diagonal bisected. d) Opposite sides are parallel and equal in length. e) Each side is the same length. Each angle is 90º.

2 Ollie describes a type of quadrilateral: ‘The diagonals bisect each other. It has less than two lines of symmetry.’ a) Write down the name of two possible quadrilaterals it could be. b) Write down another property that would help find what the shape actually is.

3 Match each description to the correct quadrilateral.

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Diagonals are equal in length and perpendicular.

Rectangle

Diagonals are perpendicular but not equal in length.

Square

Diagonals are equal in length but not perpendicular.

Parallelogram

Diagonals are not equal in length and are not perpendicular.

Kite

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3 Moira is a supermarket manager. She wants to ask customers about the freshness of the vegetables on sale in her supermarket.

She decides to carry out face-to-face interviews with customers.

She decides to select customers using one of these two methods.

Method 1: Ask 10 customers each day for one week, interviewing at different times of the day.

Method 2: Ask 80 customers using the supermarket on a Thursday evening.

Which data collection method is best? Give a reason for your answer.

4 A school has 400 boys and 100 girls. The headteacher wants to find out the views of a representative sample of students about school clubs.

She decides to write a questionnaire. She plans to choose people to answer her questionnaire using one of these three methods.

Method A

Method B

Method C

Ask for 100 volunteers to stay behind after assembly to complete the questionnaire.

Ask 80 boys and 20 girls chosen at random to complete the questionnaire.

Ask the 100 students who come first alphabetically on the school register.

Which method is best? Give reasons for your answer.

5 Gina works in a biscuit factory. She wants to check that biscuits made by a machine have the correct mass.

She can use either of the following sampling methods.

Method 1: Record the mass of every 100th biscuit made by the machine.

Method 2: Choose one packet of biscuits and find the mass of every biscuit in that packet.

Which method should she choose? Give a reason for your answer.

6 A football club wants to find out the views of fans about the price of tickets.

It considers asking 40 people attending one match. a) Suggest a better way for the club to get the information it wants. b) Explain why your method should give the club more reliable information.

Thinking and working mathematically activity Technology question A theatre has 225 seats. Every seat in the theatre is sold on one particular day. The ages of the people in the theatre are shown in the seating plan on the next page.

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Exercise 2 1 Below is a list of numbers.

1 50 −π 3 1 −88 −17.2 – 7 Write down which numbers are: a) rational numbers

7.43

68

b) integers

c) natural numbers

2 In part a) and part b), use the clues to find the number n. a) n is an integer greater than −1. It is not a natural number.

5 b) n is a natural number between and 3.85 2 3 a) Write a natural number that lies between −7 and 2 b) Write an integer that lies between −7.73 and −6.41 4 For each pair of rational numbers, find another rational number that lies between them.

1 7 a) and b) 3.1 and 4.2 c) 0.78 and 0.79 d) 0.999 and 1 4 8 5 Is each statement always, sometimes or never true?

If a statement is always true or never true, explain why.

If a statement is sometimes true, write one example where it is true and one example where it is not true. a) The product of two integers is a natural number. b) The quotient of two integers is an integer.

Tip The quotient of two numbers is the result when you divide the first number by the second number.

c) The quotient of two integers is a rational number. d) The product of two fractions is a rational number. e) The quotient of two fractions is a rational number.

6 Vocabulary question Copy the sentences below and fill in the blanks using the list of words in the box. positive

negative

rational

integers

natural

Numbers that can be written as fractions are called .................. numbers. These include positive and .................. whole numbers, which are called .................. . .................. whole numbers are called .................. numbers.

Thinking and working mathematically activity A rational number is any number that can be written as a fraction. Investigate this idea, exploring what types of number you can write as fractions.

Write a short guide to recognising rational numbers.

Draw a Venn diagram showing rational numbers, natural numbers and integers. Explain how you drew it.

How many different regions does your diagram have? Write two numbers in each region.

Chapter 4: Factors and rational numbers

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