01 ncm7 2nd ed sb txtweb2.hunterspt-h.schools.nsw.edu.au/studentshared... · 7 chapter 1 the...

345678 9012345 67890123456789012345678 90123456789012345678901234567890123456 7890123456789012 34567890123456 901234567890123 01234567890 123456789012345678901234 5678 012345678901234 5678901234567890123456789014567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 23456789012 45678901234 6789012345678901 34567890123456789012345678901234567 0123456789012345678901234567567890125678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345601234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345690123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123459012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012348901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678934567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012347890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567823456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567812345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345671234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234560123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890156789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234569012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789045678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789034567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567893456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012347890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567823456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123

234567890123456789012345678901234567890123456 7890123456789012 345678901 9012345 67890123456789012345678901234567890123456789012345678901234567890134 6789012345678901234567890123456789012345678901234567890123

456789012345678901234567890123456789012345678 56789 7890123456789012 3456789012345678 901234567890123456789012345678901234567890123456789012345567890123456789012345678901234567890123456789012345 56789012345678901234567890 123456789012345678 5678 234567890123 45678901234 0123456789

4567890123456789012345 123456789012345678 234567890123 4567890123

567890123 4567890123 8901234 0123456789

7890123 8901234 23456789

2345 31 01234 67890

1 5

4567890123 8901234

48901234

NUMBER

Mathematics began with people counting, and many civilisations came up with symbols to represent numbers. As people around the world started to cross paths, a common number system was needed. Eventually the Hindu–Arabic system was adopted all over the world. It is important to understand how our number system works and the rules it follows.

01 NCM7 2nd ed SB TXT.fm Page 2 Saturday, June 7, 2008 7:07 PM

89012345678 3456789012345678901230123

756789012345678901234901234567890123456789456789012345678901234

890123456789012345678345678901234567890123890123456789012345678

234567890123456789012789012345678901234567234567890123456789012

678901234567890123456123456789012345678901678901234567890123456

012345678901234567890567890123456789012345012345678901234567890

456789012345678901234901234567890123456789456789012345678901234

890123456789012345678345678901234567890123890123456789012345678

234567890123456789012789012345678901234567234567890123456789012

678901234567890123456123456789012345678901678901234567890123456

45678901 456789012

56789012 8 789012345678

0123456789 78901234

012345 567890123456780123456789 5678 678901234 0123456789

3 8901234

345678 9012345 67890123456789012345678 90123456789012345678901234567890123456 7890123456789012 901234567890123 01234567890 123456789012345678901234 5678 012345678901234 56789014567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 23456745678901234 6789012345678901 34567890123456789012345678901234567 012345678901234556789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345670123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345690123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789045678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234569012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678903456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789034567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123478901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567892345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012367890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456781234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234567890125678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901256789012345678901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678901234569012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678904567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234589012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678903456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123478901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567892345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234

456789

3456789012345678901234567890123456 7890123456789012 34567890 9012345 6789012345678901234567890123456789012345678901234567890123456789 6789012345678901234567890123456789012345678901234567890123

456789012345678901234567890123456789012345678 56789012 7890123456789012 3456789012345678 901234567890123456789012345678901234567890123456789012345678 567890123456789012345678901234567890123456789012345 56789012345678901234567890 1234567890123456789 5678 234567890123 45678901234 0123456789

4567890123456789012345 56789012345678 1234567890123456789 5678 234567890123 45678901234 0123456789

567890123 4567890123 8901234 0123456789

567890123 4567890123 8901234 0123456789

0123456789

48901234 0123456789

0123456789

In this chapter you will: Wordbank

• compare the Hindu–Arabic number system with number systems from different societies, past and present

• recognise, read and convert Roman numerals

• state the place value of any digit in large numbers

• order numbers of any size, in ascending and descending order

• record large numbers using expanded notation

• revise the four basic operations on whole numbers

• apply order of operations to simplify expressions

• divide two-digit and three-digit numbers by a two-digit number

• use the symbols of mathematics, including and . 3

•

cube root

The value which, if cubed, will give the required number, for example

=

4 because4

3

=

64.

•

evaluate

To find the value of a numerical expression.

•

expanded notation

A way of writing a number that shows the place value

of every digit.

•

Hindu–Arabic number system

The number system we use, with the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

•

numeral

A symbol that stands for a number, such as 8 or X.

•

order of operations

The rules for calculating an expression containing mixed

operations, such as 14

−

2

×

4

+

1.

•

place value

The way that the position of a digitin a number tells us its value.

643


4

NEW CENTURY MATHS 7

Start up

1

Write the answers to the following.

a

10

×

10

b

4

×

7

c

900

+

30

d

7

+

9

e

10

×

10

×

10

f

35

÷

5

g

9

×

9

h

26

−

8

i

1000

+

200

+

50

j

6

×

5

k

99

÷

11

l

75

−

16

m

18

×

3

n

7

×

12

o

128

−

24

p

128

÷

4

q

137

+

45

r

35

× 12

s

452

−

140

t

280

×

10

u

3601

−

59

2

Write each of these numbers in words.

a

45

b

120

c

138

d

3680

e

5001

f

47 613

3

Write each of the following numbers using numerals.

a

sixty-eight

b

seven hundred

c

two thousand and four

d

eight hundred and ninety-nine

e

ten thousand, four hundred and ninety-two

1-01 The ancient Egyptian number system

The ancient Egyptians used one of the earliest number systems about 5000 years ago. Pictures called

hieroglyphs

represented words or sounds. They were written on papyrus (a type of paper made from reeds) or painted on walls.

The hieroglyphic symbols used by the Egyptians were:

Worksheet1-01

Brainstarters 1

Worksheet1-02

Multiplication facts

Reading and writing large numbers

Skillsheet1-01

10

1 2 3 4 5 6 . . . 9

20 . . . 100

(coiled rope) (lotus flower)

200 . . . 1000

10 000 100 000

(bent reed) (fish)

1 000 000 (million)

(man with hands raised in surprise)


5

CHAPTER 1

THE HISTORY OF NUMBERS

1

If you were an ancient Egyptian student, how would you write these numerals?

a

7

b

37

c

165

d

268 301

e

3 251163

f

1253

2

Use our numerals to write the numbers represented by these Egyptian numerals.

3

Write the answer to these in Egyptian numerals.

Exercise 1-01

Example 1

Show how an ancient Egyptian would have written each of these numbers.a 25 b 126 c 3468

Solutiona b c

Example 2

If ancient Egyptian numerals could be written in any order, how could 125 be written?

Solution

or or

Ex 1

a b c

d e

plus

plus

minus

a

b

c

d

minusminusminus


6

NEW CENTURY MATHS 7

4

State one advantage and one disadvantage of working with ancient Egyptian numerals.

5

Why do you think a picture of a surprised man was used by the ancient Egyptians to represent a million?

1-02 Australian Aboriginal number systems

The Australian Aboriginal way of life had no need for a complicated number system. Their society relied on story-telling, using the spoken language rather than writing, and Aboriginal people did not have symbols for numbers. Different regions had their own names for numbers. The

Belyando River people

of central Queensland used only two words to name their numbers:1

=

wogin 2

=

booleroo3

=

booleroo wogin 4

=

booleroo boolerooThe

Kamilaroi people

lived in northern New South Wales, including the regions surrounding Moree and Tamworth. They used three words to name their numbers.1

=

mal 2

=

bularr3

=

guliba 4

=

bularr bularr5

=

bularr guliba 6

=

guliba guliba

1

How did the Belyando River people form words for the numbers 3 and 4?

2

How did the Kamilaroi people form words for 4, 5 and 6?

3

Answer the following, using the correct Aboriginal words:

a

wogin

+

booleroo wogin

b

guliba

×

bularr

c

bularr

+

bularr

+

mal

d

booleroo

×

booleroo

e

guliba guliba

−

guliba

f

bularr bularr

−

mal

4

State one advantage and one disadvantage of working with Aboriginal numbers.

1-03 The Babylonian number system

The ancient kingdom of Babylon existed from about 3000 to 200

BC

where Iraq is today. Babylonian writing used wedge shapes called

cuneiform

.

The wedges were stamped into clay tablets which were then baked. Babylonian numerals also used cuneiform.While our number system is based on 10 and 100, the Babylonian number system was based on 10 and 60. This wedge stood for 1: A sideways wedge stood for 10:

A larger wedge stood for 60:

Exercise 1-02

10 30 . . .. . .20 130120807060

1 2 3 4 5 . . . 9


7

CHAPTER 1

THE HISTORY OF NUMBERS

1 How would you write each of these numerals using our numerals?

Notice that there was no need for a zero.

2 Use Babylonian numerals to write each of these amounts.a 26 b 58 c 107d 300 e 144 f 401

3 State one advantage and one disadvantage of working with the Babylonian number system.

1-04 The Roman number systemThe Roman empire was one of the greatest empires. Roman numerals were invented about 2000 years ago. They were used until the end of the 16th century. Today they are used mainly in clocks and for some page numbers in books. The Romans used the following numerals:

1 2 3 4 5I II III IV V6 7 8 9 10

VI VII VIII IX X

50 100 500 1000L C D M

Exercise 1-03

Example 3

Show how a Babylonian would have written each of these numbers.a 15 b 252

Solutionab For numbers greater than 60, we need to find how many 60s divide into them.

252 ÷ 60 = 4 and remainder 12 because 4 × 60 = 240So 252 = (4 × 60) + 10 + 2.In Babylonian numerals, 252 is:

a b

c d

Ex 3

Worksheet1-03

Roman numerals

Skillsheet1-02

Roman numerals


8 NEW CENTURY MATHS 7

The Romans had an unusual method of writing certain numbers:• Instead of writing 4 as IIII, they wrote IV meaning V − I (that is 5 − 1 = 4).• Instead of writing 9 as VIIII they wrote IX meaning X − I (that is 10 − 1 = 9).• For 90, they wrote XC (that is 100 − 10 = 90).

1 Titus, a student in ancient Rome, wrote these numerals. Change them into our numbers.a XXVI b XL c CCLXIV d LIVe MMCLIX f MCMXC g XCVIII h MDVII

2 What would Titus have written for these numbers?a 365 b 36 c 79 d 97e 2600 f 344 g 999 h 3473

3 Why do you think Roman numerals are no longer widely used?

4 The Roman word for hundred was ‘centum’ which is why C stands for 100. List some words beginning with ‘cent’ that mean one hundred of something.

1-05 The modern Chinese number systemChinese people today use the numerals below.

• The Chinese write from top to bottom.• The symbols in a number are grouped in

pairs and the numbers in each pair are multiplied together.

• The products are added to give the number.

Exercise 1-04

Example 4

Write each of the following in Roman numerals.a 23 b 46 c 101 d 249

Solutiona 23 is XXIII b 46 is XLVI c 101 is CI d 249 is CCXLIX

Ex 4

Ancient Chinese rod numerals

Worksheet1-04

1 2 3 4 5

6 7 8 9

10 100 1000 10000

Worksheet1-05

Mayan numerals


9CHAPTER 1 THE HISTORY OF NUMBERS

1 Use our numerals to rewrite these Chinese numerals Zhang Li wrote.

2 If you were writing to Zhang Li, how would you write each of these numbers using Chinese numerals?a 13 b 46 c 175 d 999

3 What are the difficulties in working with modern Chinese numerals?

Exercise 1-05

Example 5

Write each of these Chinese numbers using our number system.

Solution

a b

3 × 100 = 300

7 × 10 = 70 +

5 = 5375

6 × 10 = 60 +

4 = 464

a b

Ex 5a b c d e

Working mathematically

Calendar monthMake a calendar for the month of your birthday using a different type of number system. Are some number systems easier to use than others? Why?

Communicating



1-06 The Hindu–Arabic number systemOur number system goes back to the Hindus (who lived in India) and came to Europe through the Middle East/Arabia. Our system needs only ten symbols called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is easier to use because it has a zero and the position of each numeral determines its value. This is called place value. The numerals first appeared in Europe in the 10th century, but were different to the ten numerals we use today.The following table shows how our numerals have changed over time.

The Hindus called the zero ‘sunya’ meaning a void. Other names used were ‘cipher’, ‘nought’ and the Arabic ‘sifr’.Even today, different cultures use different symbols:

Place valueWe can write any number using only ten symbols or digits. When we write numbers, each column has a special value called the place value.

Ancient number systems

Worksheet1-06

Hindu

Hindu

Hindu

Arabic

Spanish

Italian

Caxton(Printer)

200 BC

AD 2

AD 800

AD 900

AD 976

AD 1400

AD 1480

1 2 3 4 5 6 7 8 9 0 10OriginDate

Numerals

or or or

Skillsheet1-03

Place value

Worksheet1-07

Big numbers

Example 6

Write the value of each of the digits in 4625.

SolutionIn 4625: 5 has a value of 5 or 5 × 1

2 has a value of 20 or 2 × 106 has a value of 600 or 6 × 1004 has a value of 4000 or 4 × 1000



1 Write the value of each digit in the following numbers, then write each number in words.a 609 b 1039 c 70 104 d 504 860 e 9 134 671 f 5 837 000g 4001 h 205 689 i 34 000 036

2 Write each of the following using numerals.a eight thousand, seven hundred and ninety-sixb three million and eighty-eightc two thousand, three hundred and eighty-fived six thousand, nine hundred and sevene four hundred and twenty thousand, eight hundred and thirtyf three hundred and nine thousand, two hundred and eleveng one million, two hundred and eighty thousand, four hundred and sixtyh twelve million, nine hundred and one

Exercise 1-06

Example 7

What is the value of each of the digits in 501?

SolutionIn 501: 1 has a value of 1

0 means there are no tens (zero used to mark a place)5 has a value of 500

Another way to show the meaning of each digit in a number is with a place-value table.

Ten thousands Thousands Hundreds Tens Ones

1 3 8 138

4 6 2 5 4625

5 0 1 501

8 2 3 5 0 82 350

Example 8

What value does the digit 5 have in:a 57? b 235?

Solutiona In 57, the 5 has a value of 50 (or 5 tens).b In 235, the 5 has a value of 5 (or 5 units).

Ex 6



3 What are the advantages of using a Hindu–Arabic number system?

4 What is the value of 7 in 237 601? Select A, B, C or D.A 7 hundred B 7 thousandC 70 thousand D 7 hundred thousand

5 In 2 982 645, which digit is in the ten thousands place? Select A, B, C or D.A 2 B 9 C 8 D 6

6 Place these numbers in a place-value table, as shown on the previous page.a 48 b 382 c 2751d 3020 e 15 364 f 44 040

7 What is the value of the digit 5 in each of these numbers?a 45 b 1057 c 1526d 12 345 e 65 013 f 51 480 260

8 What is the value of the digit 3 in each of these numbers?a 123 b 2356 c 32 185d 85 532 e 1 385 264 f 3 485 260

9 What is the value of the digit 4 in each of these numbers?a 4281 b 124 386 c 6004d 4 316 725 e 362 154 f 1 426 813

10 Arrange the numbers in each of these sets in order, from smallest to largest.a 321, 17, 8000 b 17, 707, 27, 63c 246, 3596, 5369, 432, 16, 6125 d 123, 321, 132, 231, 213e 1045, 450, 145, 82 f 721, 243, 43, 4372, 722g 380 211, 308 022, 300 806, 392 084 h 4 856 231, 4 766 372, 1 429 950, 3 006 853

11 How many times is the first 3 bigger than the second 3 in each of these numbers?a 1433 b 1343 c 3143 d 2 352 312

1-07 Expanded notationOne way to show the place value of each digit in a number is to use expanded notation.

Ex 7

Ex 8

Base 8 number system

Worksheet1-08

Example 9

Write each of these numbers using expanded notation.a 345 b 3287

Solutiona 345 = (3 × 100) + (4 × 10) + (5 × 1)

= 3 × 102 + 4 × 10 + 5 × 1b 3287= (3 × 1000) + (2 × 100) + (8 × 10) + (7 × 1)

= 3 × 103 + 2 × 102 + 8 × 10 + 7 × 1



1 Write each of these numbers using expanded notation.a 56 b 3562 c 416 d 502 e 1001f 10 253 g 38 002 h 59 644 i 3809 j 120 435

2 Write each of these as a single number.a (5 × 100) + (2 × 10) + (4 × 1)b (6 × 1000) + (5 × 100) + (3 × 10) + (7 × 1)c (4 × 102) + (2 × 10) + (9 × 1)d (6 × 103) + (4 × 102) + (7 × 10) + (3 × 1)e 8 × 104 + 2 × 103 + 3 × 102 + 4 × 10 + 3 × 1f 3 × 103 + 0 × 102 + 5 × 10 + 7 × 1g 7 × 104 + 6 × 103 + 0 × 102 + 0 × 10 + 1 × 1h 1 × 104 + 0 × 103 + 9 × 102 + 9 × 10 + 9 × 1i 3 × 105 + 4 × 104 + 4 × 103 + 2 × 102 + 2 × 10 + 0 × 1j 9 × 105 + 0 × 104 + 0 × 103 + 9 × 102 + 9 × 10 + 9 × 1

3 What is 9047 in expanded notation? Select A, B, C or D.A 9 × 1000 + 4 × 100 + 7 × 10 B 9 × 1000 + 4 × 10 + 7 × 1C 9 × 1000 + 4 × 100 + 7 × 1 D 9 × 100 + 4 × 10 + 7 × 1

4 Find out what ‘to expand’ means. Is the dictionary meaning the same as the one in mathematics?

Exercise 1-07

102 10 squared means 10 × 10 = 100103 10 cubed means 10 × 10 × 10 = 1000104 10 to the power of 4 means 10 × 10 × 10 × 10 = 10 000

The power of 10 shows how many zeros follow the 1 in the number.

!

Ex 9

Just for the record

Googol-plexingThe number 10100, the googol, is 1 followed by one hundred zeros. The name ‘googol’ was created by the 9-year-old nephew of American mathematician Dr Edward Kasner.The number 10googol, that is 1 followed by a googol zeros, is called the googolplex.The googol is a very big number but it is rarely used for practical purposes. Even the number of particles in the observable universe, estimated at being between 1072 and 1087, is less than a googol!The Internet search engine Google was named after the googol, to reflect the huge size of the world wide web. It was invented in 1996 by two Stanford University students, Larry Page and Sergey Brin. Google is a powerful search engine becauseit can find information from at least 25 billion web pages in less than 1 second.How many googols are there in a googolplex?



1-08 The four operationsThere are four basic operations in our number system:

+ addition × multiplication − subtraction ÷ divisionThe old symbols for writing these operations are:

We will now review these operations.

Mental skills 1A

Multiplying by a multiple of 10Place value allows us to simply add zeros to the end of a number whenever we multiplyby a power of 10. The zeros at the end shift all the other digits one or more places to the left which results in them having higher place values.

1 Examine these examples.a 37 × 10 = 370 b 45 × 100 = 4500c 16 × 1000 = 16 000 d 100 × 1000 = 100 000e 7 × 90 = 7 × 9 × 10 = 63 × 10 = 630f 5 × 400 = 5 × 4 × 100 = 20 × 100 = 2000g 12 × 300 = 12 × 3 × 100 = 36 × 100 = 3600h 40 × 800 = 4 × 10 × 8 × 100 = 4 × 8 × 10 × 100 = 32 × 100 = 32 000

2 Now simplify these.a 18 × 100 b 26 × 1000 c 77 × 10 000 d 10 × 100e 315 × 1000 f 1000 × 1000 g 294 × 10 h 475 × 100i 3 × 80 j 8 × 200 k 6 × 50 l 7 × 30m 2 × 6000 n 11 × 900 o 4 × 400 p 5 × 700q 5 × 80 r 25 × 20 s 300 × 60 t 900 × 4000

Maths without calculators

Worksheet1-09

Four operations

Example 10

Copy and complete this number grid.

Solution

+ 5 14

8

12

+ 5 14

8

12

+ 5 14

8 13 22

12 17 26

+ 5 14

8 13 22

12 17 26

5 + 8 14 + 8

14 + 125 + 12



Use the link to Worksheet 1–10 to print the number grids in this exercise.1 Copy and complete these number grids.

2 Copy and complete these number grids.a top row − side column b top row − side column c top row − side column

3 Copy and complete these number grids.

Exercise 1-08

Example 11

Copy and complete this number grid.

Solution

+ 12

30

20 65

+ 12

30

20 65

+ 12 45

18 30 63

20 32 65

+ 12 45

18 30 63

20 32 65

30 − 12 18 + 45

65 − 20

20 + 12

Worksheet1-10

Number grids

Ex 10

Worksheet1-11

Arithmagons

a b c+ 3 4

7

2

+ 15 41

28

19

+ 11 9

8

5

The take-away bar: go figure

L 102TLF

The multiplier: go figure

L 90TLF− 19 25

7

12

− 54 78

37

26

− 243 412

128

239

The divider: with or without remainders

L 2006TLF

The multiplier: make your own hard multiplications

L 82TLFa b c× 11 9

8

5

× 2 5

15

23

× 12 20

10

17



4 Copy and complete these number grids.a top row ÷ side column b top row ÷ side column c top row ÷ side column

5 Copy and complete these number grids.

6 Find the missing numbers (top row − side column).

7 Find the missing numbers.

8 Find the missing numbers (top row ÷ side column).

÷ 36 48

4

3

÷ 32 64

8

4

÷ 60 100

4

5

Ex 11

a b c+ 10

50

80 100

+ 16

26 28

13

+

22 33

6 14

a b c− 20 15

8

9

− 17

9 15

11

−

7 9 12

11

a b c× 5

3 12

28

× 5

56 40

7

× 10 6

90

4

a b c÷ 24

6 3

4

÷

8 4

2 24

÷ 72

24 10

5



Using technology

What is a spreadsheet?A spreadsheet is like a calculator. We can enter data and solve many problems more easily, using an Excel spreadsheet.Spreadsheets are made up of many cells. As we go across the page, we change the column (A, B, C, D, etc.). As we go down the page, the row changes (1, 2, 3, 4, etc.).

Using formulasTo write a formula in a cell, always start with an equal sign ‘=’. A spreadsheet uses special symbols to do calculations. Consider these basic operations:

a =A1+A2+A3 or =sum(A1:A3) means add the values in cells A1, A2 and A3b =A5-A4 means subtract the value in cell A4 from the value in cell A5c =A1*A3 means multiply the value in cell A1 by the value in cell A3d =A5/4 means divide the value in cell A5 by 4 (/ is used instead of ÷)e =A2^2 means square the value in cell A2 (instead of (A2)2)f =average(A1:A5) means find the average of all values in cells A1 to A5

1 Enter the following numbers into cells as shown below, where m represents the value in cell B1, n is the value in cell B2, p is the value in cell B3, and so on.

2 Enter the following formulas into the given cells.

a C1, q − 7 (means enter the formula into cell C1 as shown above)b C2, n − m c C3, 2 × r − 7d C4, 3 × ( p + q) e C5, p × q × r

f C6, p2 g C7,

h C8, i C9, m + n + p + q + r

j C10, average of m, n, p, q and r k C11, −

3 Choose different values and enter them into cells B1 to B5. Consider the new answers obtained in column C, for the formulas entered from question 2.

3 m×2

--------------

r p–3

-----------qm---- r

p----

Skillsheet1-04

Spreadsheets




Double-digit dice gameThis is a game for two or more players using one die. InstructionsStep 1: Copy the scoresheet shown on the right.Step 2: Each player rolls the die seven times and,

for each roll, can choose to write the number in either the tens column or the units column of his or her scoresheet.

Step 3: Each player finds the total of his or her seven numbers. The winner is the person with a total closest to 99.

Step 4: Play the game again and work out a strategy to improve your score.

Scoresheet

Roll Tens Units

1st

2nd

3rd

4th

5th

6th

7th

Total

Applying strategies and reasoning

Mental skills 1B

Dividing by a multiple of 10Place value allows us to remove zeros from the end of a number when we divide by a power of 10. The deleted zeros shift all the other digits one or more places to the right which results in them having lower place values.

1 Examine these examples.a 2000 ÷ 10 = 2000 ÷ 10 = 200b 1800 ÷ 100 = 1800 ÷ 100 = 18c 37 000 ÷ 100 = 37 000 ÷ 100 = 370d 6 000 000 ÷ 1000 = 6 000 000 ÷ 1000 = 6000e 6000 ÷ 200 = 6000 ÷ 100 ÷ 2 = 60 ÷ 2 = 30f 350 ÷ 70 = 350 ÷ 10 ÷ 7 = 35 ÷ 7 = 5g 2800 ÷ 40 = 2800 ÷ 10 ÷ 4 = 280 ÷ 4 = 70h 40 000 ÷ 5000 = 40 000 ÷ 1000 ÷ 5 = 40 ÷ 5 = 8

2 Now simplify these.a 200 ÷ 10 b 6000 ÷ 100 c 45 000 ÷ 100 d 30 000 ÷ 1000e 1900 ÷ 10 f 2600 ÷ 100 g 530 ÷ 10 h 720 000 ÷ 1000i 180 ÷ 30 j 300 ÷ 50 k 1600 ÷ 400 l 45 000 ÷ 5000m 4200 ÷ 60 n 21 000 ÷ 700 o 44 000 ÷ 2000 p 1600 ÷ 200q 24 000 ÷ 600 r 24 000 ÷ 3000 s 64 000 ÷ 80 t 5400 ÷ 900

Maths without calculators



1-09 Dividing by a two-digit numberIn primary school, you studied division by a single-digit number. We will now divide numbers by a two-digit number using two different methods.

1 Find the answers for the following.a 180 ÷ 15 b 462 ÷ 22 c 731 ÷ 17d 666 ÷ 18 e 992 ÷ 31 f 78 ÷ 13 g 900 ÷ 25 h 667 ÷ 23 i 85 ÷ 17

Exercise 1-09

Worksheet1-09

Four operations

Example 12

Divide $312 among 12 people.

SolutionMethod 1: Long division Method 2: Preferred multiples

Each person receives $26.

2612 312 12 into 31 is 2

−24 ↓72 12 into 72 is 6

−720

2612 312

−120 10 times192

−120 10 times72

−72 6 times0 26 times

Example 13

Simplify 296 ÷ 21. Then complete: 296 = × + .

SolutionMethod 1: Long division Method 2: Preferred multiples

Answer = 14 , so 296 = 14 × 21 + 2

14 remainder 221 296 21 into 29 is 1

−21↓86 21 into 86 is 4

−842

14 remainder 221 296

−210 10 times86

−84 4 times2 14 times

221------

Ex 12



2 Carry out these divisions and write your answers in the form: = × + .a 304 ÷ 12 b 505 ÷ 14 c 99 ÷ 26d 917 ÷ 19 e 958 ÷ 34 f 869 ÷ 28 g 594 ÷ 27 h 79 ÷ 13 i 815 ÷ 40

3 At a party 275 lollies are shared equally among 25 children. How many lollies does each child get?

4 A piece of wood 390 cm in length is to be cut into 15 equal pieces. How long is each piece?

Ex 13


Magic squaresMagic squares have every row, column and diagonal adding to the same magic sum. The Lo-Shu magic square dates back to about 2200 BC. It appeared on an ancient Chinese tablet and was first drawn on a tortoise shell given to the Emperor Yu.

Reasoning

1 a Draw a 3 × 3 magic square frame. Write the Lo-Shu magic square into your frame using the numbers 1 to 9. (Hint: Count the dots. Top left-hand corner is a 4.)

b What is the magic sum for the Lo-Shu square?

2 Which of these squares are not magic?

3 Make these squares magic by finding the missing numbers.

42 14 34

22 30 38

26 46 18

21 0 15

12 6 18

3 30 5

a b c 38 8 28

16 24 32

20 30 12

29 19 33

21 35

44

39 49

34

a b c 21 6

12 45

48 27

33 3 42

Worksheet1-12

Magic squares



4 Another famous magic square appears in a woodcut by the German artist Albrecht Dürer, who lived from 1471 to 1528. It is called the magic square of Jupiter.

a Find the 4-digit numeral contained within the square that identifies a year that occurred during Dürer’s lifetime.

b What is the magic sum for this 4 × 4 square?

c Find five 2 × 2 squares within the magic square for which the numbers have the same total as the magic sum.

d Apart from the two diagonals, find four numbers each from a different row and column that add to the magic sum. There are more than two solutions.

Using technology

Sorting dataSort the set of numbers {60, 107, 85, 6, 28, 45, 265} using a spreadsheet, by following the instructions shown below.

1 a Enter the numbers, in the given order, into column A.

b Highlight cells A1 to A8 and choose Data and Sort.



1-10 Order of operations

c Choose Sort by Column A and Ascending as shown below.

d The data should now be sorted from smallest number (cell A1) to the largest number (cell A7).

e A set of numbers can also be sorted in descending order. Highlight the cells and choose Sort by Column A and Descending.

2 Now sort these sets of numbers in the columns given, by repeating this method.a Enter {55, 89, 36, 21, 19, 4, 95} in column Bb Enter {263, 141, 940, 508, 836, 392, 1063} in column Cc Enter {4987, 4200, 8740, 9005, 2601, 2514, 4810} in column Dd Enter {16 101, 12 167, 10 010, 11 412, 10 107, 10 761, 11 214} in column E

3 Sort the data from question 2 in descending order, for each of columns B to E.

The order of operations rulesFirst: Work out the value within any grouping symbols, starting with the innermost

grouping symbols: parentheses or round brackets ( ) square brackets [ ] braces { }.

Second: Work out multiplication or division as you come to it, going from left to right.Third: Work out addition or subtraction as you come to it, going from left to right.

!

Example 14

Find the value of (5 + 13) ÷ 2.

Solution(5 + 13) ÷ 2 work out grouping symbols

= 18 ÷ 2 division

= 9 answer

{{



1 Evaluate (find the value of) each of the following.a 12 × (3 + 5) b (16 − 3) × 2 c (60 + 12) ÷ 6d (3 − 2) × 5 e (2 + 5) × 6 f (12 − 4) ÷ 4 g 7 × (25 − 12) h 36 ÷ (14 − 10) i (5 × 7) − 16j 120 ÷ (34 − 24) k 5 + 6 × (50 − 10) l (77 ÷ 11) − 7

2 Evaluate the following.a 3 + 5 × 2 b 20 − 2 × 5 c 5 + 3 × 2 − 7d 19 − 4 × 4 − 1 e 24 − 5 ÷ 5 + 7 f 17 + 8 − 3 × 2g 2 × 10 − 9 + 28 h 42 ÷ 7 − 5 i 9 + 28 − 12j 4 × 8 − 3 × 3 k 109 + 36 ÷ 4 l 60 − 8 × 4 + 20

3 12 ÷ 4 + 8 × 5 = ? Select A, B, C or D.A 5 B 16 C 43 D 55

4 Find the answer to each of the following.a (24 − 4) ÷ 5 + 7 b 2 × (10 − 9) + 28 c (8 + 2) × (17 − 7)d 7 + 7 + (11 − 8) e (16 − 5 + 8) × 9 f (8 + 8 − 5) × (7 + 4)g 9 + 3 × (15 − 4) − 5 × 6 h 16 × 3 − 4 × (15 − 6 × 2) + 7i (5 + 8) × 2 − (25 ÷ 5) j 4 × [(5 + 11) ÷ 2] − (15 × 2)

Exercise 1-10

Example 15

1 Find the value of 15 ÷ 5 × 8.

Solution15 ÷ 5 × 8 division

= 3 × 8 multiplication

= 24 answer

{{

2 Find the value of 5 + 6 × 2 − 7.

Solution5 + 6 × 2 − 7 multiplication

= 5 + 12 − 7 addition

= 17 − 7 subtraction

= 10 answer

{{

{

Example 16

Find the value of 25 − [7 × (5 − 3) + 4].

Solution25 − [7 × (5 − 3) + 4] innermost grouping symbols

= 25 − [7 × 2 + 4] grouping symbols: inside multiplication first

= 25 − [14 + 4] grouping symbols

= 25 − 18 subtraction

= 7 answer

{{

{{

Ex 14

Ex 15

Ex 16



k 100 − [12 + (3 × 5) ÷ 3]l 120 ÷ {16 + [(2 × 5) + 4]}m {15 − [3 × (12 − 9) + 1]} − [(44 × 2) + 12] ÷ 50n [(16 − 4) × 10] ÷ [(45 ÷ 3) + 25]o 86 + [(15 ÷ 3) + (65 ÷ 5)] × 2 p [20 ÷ (5 − 4) × 2] − {[(4 + 5) × 3] ÷ [15 − (30 ÷ 5)]}

5 Put grouping symbols where necessary to make each of the following statements true. The first one has been done for you.a 5 − 2 × 4 = 12 becomes (5 − 2) × 4 = 12 b 3 + 8 − 7 = 4c 15 − 3 × 5 = 60 d 15 − 3 × 5 = 0 e 8 + 4 − 3 × 2 = 10f 8 + 4 − 3 × 2 = 6 g 8 + 4 − 3 × 2 = 18 h 6 + 4 × 0 = 6i 6 + 4 × 0 = 0 j 100 ÷ 10 + 10 = 5 k 100 ÷ 10 + 10 = 20

6 Put grouping symbols where necessary to make each of the answers correct.a 84 ÷ 3 + 9 × 15 − 11 = 152 b 84 ÷ 3 + 9 × 15 − 11 = 64c 84 ÷ 3 + 9 × 15 − 11 = 94

7 Use the four numbers in each set only once (in any order), with the operations +, −, ×, ÷ or grouping symbols, to make an equation that equals the number in the red box.

a 2, 7, 8, 9 b 1, 2, 3, 5 c 3, 4, 6, 8

d 2, 6, 8, 1 e 2, 4, 6, 8 f 2, 5, 8, 10

g 3, 5, 7, 9 h 4, 5, 7, 9 i 2, 5, 7, 10

1-11 The symbols of mathematicsMathematics does not only involve numbers. It has a language of its own and uses symbols recognisable throughout the world. This table shows some of the most common symbols.

The square root of a given number is the positive value which if squared will give that number. The cube root of a number is the value which if cubed will give the number.

Symbol Meaning Symbol Meaning

+ plus, add, sum square root ( = 5)

− minus, subtract, difference cube root ( = 2)

× multiply, times, product ∴ therefore

÷ divided by, quotient � or � approximately equal to

= equal to 32 squared (3 × 3)

≠ not equal to 53 cubed (5 × 5 × 5)

� less than ( ) parentheses or brackets

� less than or equal to [ ] square brackets

� greater than { } braces

� greater than or equal to

12 18 41

21 10 44

2 8 60

Cross number puzzle

Worksheet1-13

25

3 83



1 Here is a list of words that relates to the four basic operations +, −, × and ÷.plus minus times multiply and dividesubtract share decrease product difference lessincrease total lots of quotient take away more than

Draw a table with column headings as shown below in your notebook, and write each of the given words in the appropriate column.

2 Rewrite these questions using mathematical symbols.a 15 minus 6 b 48 plus 12 c 12 is greater than 5d 5 is not equal to 3 plus 6 e the product of 7 and 8 f the square root of 16g 36 divided by 4 h 5 squared i 8 more than 12j 6 less than 13 k increase 3 by 13 l the quotient of 39 and 3m the difference between 25 and 8 n the cube root of 125o 13 is not equal to 3 p 999 is approximately equal to 1000

3 Write the answer to each of the following.a the number 6 less than 18 b the sum of 26 and 14 c the total of 6, 8 and 22d 9 times 8 e 7 squared f the quotient of 36 and 4g the number 14 more than 8 h decrease 33 by 11 i increase 83 by 27j 7 lots of 13 k the cube root of 64l the difference between 135 and 29

Exercise 1-11

+ − × ÷

Example 17

Find the answer for each of the following.a 62 b c

Solutiona 62 = 6 squared = 6 × 6 b = the square root of 9

= 36 = 3 since 32 = 3 × 3 = 9

c = the cube root of 125= 5 since 53 = 5 × 5 × 5 = 125

9 1253

9

1253

Example 18

Write the meaning of each of the following.a 3 � 7 b 5 � 5

Solutiona 3 is less than or equal to 7. b 5 is greater than or equal to 5.



4 Which of these statements is true? Select from A, B, C or D.

A = 18 B 18 ÷ 2 ≠ 9 C 6 × 4 � 15 D 72 � 12

5 Write whether each of the following is true (T) or false (F).a 16 � 2 b 42 = 8 c 300 � 5 × 100

d 3602 = 3600 e = 5 f 8 × 201 � 8 × 200g 2 � h product of 2 and 15 = 17 i 63 ÷ 3 � 60 ÷ 5j 33 = 27 k 52 − 3 = 7 l 72 � 73

m 16 × 0 � 7 × 0 n (30 − 6) × 5 � 12 × 10 o = 6

p = 1 q 53 = 15 r � 4

6 Complete the blank with � or � to make each statement true.a 7130 860 b 2001 2010c 352 140 4 082 716 d 2651 2561e 3602 3206 f 13 253 1353g 8079 8097 h 1432 1483

7 For each of the following statements, select all the numbers from this list of seven numbers that make the statement true: 2, 3, 7, 8, 11, 36, 41.a � 13 b � 5 c � 8 d � 42

e 3 = 8 f � 11 g = 2 h 5 + � 8

36

Ex 17

25273

36

13 24

Ex 18

3


The four 4s puzzleForm 10 groups (Group A, Group B, Group C, etc.). Use only four 4s and any of the mathematical symbols =, −, ×, ÷, brackets, a decimal point (.), factorial (!) or square root ( ) to make expressions for all the numbers from 1 to 100. Group A doesthe numbers 1 to 10, Group B does 11 to 20, … Group J does 91 to 100.Here are some suggestions:• 4 + 4 × 4 + 4 = 4 + 16 + 4 = 24 • 4 × 4 − 4 ÷ 4 = 16 − 1 = 15• 4! + 4 × 4 ÷ 4 = 24 + 4 = 28 • 4 × 4 + 4 × 4 = 16 + 16 = 32

(Hint: 4! = 4 × 3 × 2 × 1)

Brain benderVarious forms of ‘brain benders’ are common in daily newspapers and magazines. Here is one for you. Copy the grids and fill in the six gaps to complete each of the lines, using the remaining digits from 1 to 9 only once. Be sure to use the ‘order of operations’ rules. The aim is to make the sum of the answers for the three lines total 45.

5 + × =

× 3 − =

− + 4 =

45

Applying strategies and reflecting



Using technology

Fruit pickingAn orchardist employed people to pick fruit in his orchard over the summer. The table below shows the types of fruit grown and numbers of bins of fruit picked each day in a particular week.

1 Copy the table, as shown, into a spreadsheet.

2 To find the total number of bins of fruit picked on Monday, type the formula =sum(B2:B5) in cell B6.

3 To copy this formula into cells C6 to F6, click on cell B6 and Fill Right by grabbing the bottom right-hand corner of the cell and dragging across to cell F6. Let go of the mouse and you will see the totals for each day.

4 Use the sum formula in cell G2 to find the number of bins of apples picked in this particular week. Use Fill Down to copy the formula into cells G3 to G6. Centre the totals calculated in the ‘G’ column.

5 Answer the following questions in the given cell. In cell:a A8, type the number of bins of fruit pieces picked on Wednesdayb A9, write a formula to find how many more bins of oranges than apples were

picked in this week.c A10, write a formula to find how much more fruit was picked on Wednesday

compared to Monday in this week.d A11, write a formula to find how many bins of lemons and mandarins in total

were picked in this week.e A12, write the day of the week on which the most fruit was picked.f A13, write the day of the week on which the least fruit was picked.g A14, type the total number of bins of fruit picked in this particular week.

Skillsheet1-04

Spreadsheets



Power plus

Cryptic arithmeticSimple codes can be made by replacing letters with other letters, symbols or numbers. Number codes are studied in a branch of mathematics called cryptic arithmetic. Your challenge is to figure out which letter replaces which number.The addition: 99 could become: KK

+ 22 + DD

121 RDRwhere K = 9, D = 2 and R = 1.Note that K + D gives an answer bigger than 10 so carrying will be involved. To solve cryptic arithmetic problems, you need to know about carrying digits when adding. Choose any of the following problems from 1 to 7.1 ON + ON + ON + ON = GO Hint: Set it out as a column sum.

2 N I NE Hint: Try R = 0 and N = 5−F OUR

F I VEThere are 71 other possible solutions. In many of these (but not all) R = 0 and N = 5. Can you find two other solutions? How many different solutions can the class find?

3 FORT Y Hint: T = 8 and Y = 6T EN

+ T EN

S I XTYThe key to this problem is to decide what value is N + N and what value is E + E.

4 THRE E+ FOUR

S EVENFor this puzzle there are 38 possible solutions.Hint: Try E = 6 and V = 0 for one solution. Try E = 5 and V = 1 for another solution. Try H = 9 and R = 4 for another.How many different solutions can the class find?

5 On a holiday, Carlos ran short of money. He sent an email to his parents:S END

+MOR E

MONEYThe value of ‘MONEY’ is the amount Carlos asked for. If Carlos asked for more than $10 000 and less than $20 000, find out how much money he asked for.

6 a RE AD b READ+ TH I S − TH I S

P AGE P AGEThese are two different problems, so R and the other letters have a different value in each problem.

7 Try to create a cryptic arithmetic question of your own. (It is not as easy as it seems!)



Chapter 1 review

Language of mathsbraces cube root difference digitevaluate expanded notation grouping symbols Hindu–Arabiclong division million number system numeralorder of operations parentheses place value preferred multiplesproduct quotient square brackets square rootsum

1 What is ‘expanded notation’? Explain in your own words.

2 What is a thousand thousands?

3 What is the Roman numeral for 500?

4 Write and name the three types of grouping symbols.

5 With which arithmetic operation would you associate the word:a quotient? b difference?

6 What is the meaning of each of these symbols?a � b

Topic overview• In your own words, write what you have learnt about the history of numbers.• Is there anything you did not understand? Ask a friend or your teacher for help.• Copy this overview into your workbook and complete it using what you have learnt in

this chapter. Ask your teacher to check your overview.

Number find-a-word

Worksheet1-14

3

Order of operations•••

Four operations••••

Hindu–Arabic numerals0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Place value•••

Symbols• +, −, ×, ÷• , ••

3

Early number systems• Egyptian• Aboriginal•••

HISTORY

OFNUMBERS



Chapter revision1 Write these using Egyptian numerals.

a 13 b 2402

2 Write these using the words of the Kamilaroi Aboriginal people.a 3 b 5

3 Write these using Babylonian numerals.a 32 b 110

4 Write each of the following in Roman numerals.a 12 b 40c 179 d 2004

5 Write these using modern Chinese numerals.a 17 b 82

6 Write each of the following using numerals.a six hundred and twelveb nine hundred and forty-threec five thousand, four hundred and ninety-nined six thousand and twoe nine million, seven hundred and fifty thousand and seventy-six

7 Arrange the numbers in each of these sets in order, from largest to smallest.a 16, 21, 38, 19, 14b 89, 36, 101, 98, 88c 2356, 2534, 2635, 2300, 2533d 12 391, 12 913, 11 990, 11 391, 12 300

8 What is the place value of the digit 4 in:a 47? b 3024?c 8412? d 146 235?

9 Write each of these using expanded notation.a 19 b 283c 665 d 42 891

10 Find the answers to these.a 36 + 58 b 127 + 81c 39 − 17 d 78 − 39e 2501 + 58 f 26 × 9g 123 × 5 h 36 × 11i 36 ÷ 4 j 252 ÷ 7k 750 ÷ 6 l 3500 ÷ 10

Exercise 1-01

Exercise 1-02

Exercise 1-03

Exercise 1-04

Exercise 1-05

Exercise 1-06

Exercise 1-06

Exercise 1-06

Exercise 1-07

Exercise 1-08

Topic test 1



11 Find the answers to these. Write your answer in the form: = × + .

a 384 ÷ 16 b 912 ÷ 19c 784 ÷ 17 d 877 ÷ 23

12 Find the value of each of these.a 16 − (5 × 3) b 6 + 5 × 3c 30 − 10 ÷ 2 d (16 ÷ 2) + (18 − 11)e (320 − 120) × 12 f 35 × (19 − 17) × 20g (36 − 14) × 2 ÷ 4 h 36 − (28 − 13) + (20 − 3 × 5)i (256 − 120) ÷ 17 j [394 + (30 ÷ 5)] ÷ (440 ÷ 11)k 36 − (4 × 3) ÷ (35 − 23) l 2 000 000 − [(300 × 100) + 1]

13 Use ‘order of operations’ to calculate:a 12 + 7 − 2 × 3 b 15 − 2 × 4 + 6 ÷ (8 − 5)c 24 + 16 ÷ 4 × 16 − 4 + 9 d 15 + (64 + 2) ÷ 3 − 16e 18 + 6 ÷ 3 − 3 + 2 × 5 f 166 + 12 × 3 − 48 ÷ 4

14 Use grouping symbols and operations signs (+, −, ×, ÷) to make each of these true.a 7 ? 3 ? 1 = 9 b 10 ? 5 ? 5 = 10c 8 ? 3 ? 6 ? 2 = 8 d 28 ? 4 ? 7 = 49e 6 ? 4 ? 3 ? 5 = 40 f 19 ? 1 ? 5 ? 3 ? 1 = 0

15 Write whether each of these is true (T) or false (F).a 5 � 8 b 7 � 2 + 4c 52 �10 d 6 × 7 � 43

e 23 � 5 + 1 f = 6

Exercise 1-09

Exercise 1-10

Exercise 1-10

Exercise 1-10

Exercise 1-11

36


01 ncm7 2nd ed sb txtweb2.hunterspt-h.schools.nsw.edu.au/studentshared... · 7 chapter 1 the...

Documents