Chapter 3

03 NCM7 2nd ed SB TXT.fm Page 76 Saturday, June 7, 2008 3:51 PM

NUMBER

Numbers form many interesting patterns. You already know about odd and even numbers. Pascal’s triangle is a number pattern that looks like a triangle and contains number patterns. Fibonacci numbers are found in many living things: the number of petals on a flower will be a Fibonacci number. You will learn how these different patterns are formed to help you to understand how numbers behave.

3

1234

48901234

4567890123 8901234 5 567890123

4567890123 8901234 0123456789 7890123 8901234 23456789 4567890123456789012345 123456789012345678 234567890123 4567890123

456789012345678901234567890123456789012345678 56789 7890123456789012 3456789012345678 901234567890123456789012345678901234567890123456789012345 567890123456789012345678901234567890123456789012345 56789012345678901234567890 123456789012345678 5678 234567890123 45678901234 0123456789 234567890123456789012345678901234567890123456 7890123456789012 2345 3456789014 9012345 678901234567890123456789012345678901234567890123456789012345678901 34 6789012345678901234567890123456789012345678901234567890123 234 6789012 345678 9012345 67890123456789012345678 90123456789012345678901234567890123456 7890123456789012 34567890123456 901234567890123 01234567890 123456789012345678901234 5678 012345678901234 567890123456789012345678901 4567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 23456789012 45678901234 6789012345678901 34567890123456789012345678901234567 012345678901234567890123456756789012 56789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 01234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901 567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456 90123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 45678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 9012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 89012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234 78901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678 23456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678 12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 67890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 1234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012 56789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 01234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901 56789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 90123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 45678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 89012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 3456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 78901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678 23456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 78901234567890 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678

03 NCM7 2nd ed SB TXT.fm Page 77 Saturday, June 7, 2008 3:51 PM

0123456789

48901234 0123456789

0123456789 567890123 3 8901234 456789 012345 0123456789 678901234

4567890123 8901234 0123456789

In this chapter you will: 56789012345678 5678

567890123

4567890123 8901234 Wordbank 0123456789 4567890123456789012345

56789012345678

• composite number A number • identify special groups of numbers: triangular, 1234567890123456789 5678 with more than 0123456789 square, Fibonacci, Pascal’s triangle and 234567890123 45678901234 two factors. 0123456789 56789012 456789012345678901234567890123456789012345678 56789012 8 7890123456789012 3456789012345678 palindromes • divisibility test A rule for testing whether a 789012345678 901234567890123456789012345678901234567890123456789012345678 • test567890123456789012345678901234567890123456789012345 numbers for divisibility number is divisible by a specific value, for 0123456789 56789012345678901234567890 1234567890123456789 example, divisible by 3. • identify the factors of a number and distinguish 78901234 5678 234567890123 45678901234 0123456789 between prime and composite numbers • factor A value that divides evenly into a given 45678901456789012345678901234567890123456 7890123456789012 345678901 number, for example, 3 is a factor of 15. 456789012 9012345 678901234567890123456789012345678901234567890123456789012345678901 • find the highest common factor of two or more 6789012345678901234567890123456789012345678901234567890123 numbers • tree A diagram that lists the prime factors 89012345678 345678 9012345 67890123456789012345678 factor 90123456789012345678901234567890123456 7890123456789012 345678901234567890123 901234567890123 01234567890 123456789012345678901234 5678 012345678901234 5678901 of a number. • express a number as a product of its prime factors 0123 4567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 234567 • index notation Using powers to write the • calculate squares6789012345678901 and cubes 756789012345678901234 45678901234 34567890123456789012345678901234567 0123456789012345 repeated multiplication of a number, for 901234567890123456789 5678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 • estimate and calculate square roots and cube 4567890123456789012340123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 example, 35. roots 890123456789012345678 567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456 • palindrome A number or word that reads the 345678901234567890123901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 • find4567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 square roots and cube roots of numbers 890123456789012345678 same forward and backward, for example, ‘2002’,12345678901234567890 234567890123456789012 9012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 expressed as a product of their prime factors. 7890123456789012345673456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 and ‘madam’. 2345678901234567890128901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 • prime number A number with only1234567890123456789012345678901234 two factors, 678901234567890123456 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789017890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 1 and the number itself. 6789012345678901234562345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 012345678901234567890 789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678 5678901234567890123451234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 0123456789012345678906789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678 456789012345678901234 1234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012 9012345678901234567895678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 4567890123456789012340123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 890123456789012345678 56789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456 345678901234567890123901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 8901234567890123456784567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 234567890123456789012 901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 7890123456789012345673456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 2345678901234567890128901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 678901234567890123456 3456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 1234567890123456789017890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 6789012345678901234562345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 012345678901234567890 78901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678

03 NCM7 2nd ed SB TXT.fm Page 78 Saturday, June 7, 2008 3:51 PM

Start up Worksheet 3-01 Brainstarters 3

Skillsheet 3-01 Classifying whole numbers

1 List the first ten even numbers. 2 Sort these numbers, putting all the even numbers in one group, and the odd numbers in another: 17 2002 371 134 60 023 2 748 691 90 704 006 1 95 13 2074 1 000 000 99 999 1256 3 Find all the numbers that divide into 6. 4 Find all the numbers that divide into 24. 5 Find all the even numbers that divide into 36. 6 Find all the odd numbers that divide into 90. 7 How can you tell if a number is even without dividing it? 8 How can you recognise an odd number? 9 Write the next three numbers in each of these patterns: a 8, 10, 12,

,

,

c 101, 103, 105,

,

e 44, 39, 34,

,

,

b 27, 30, 33,

,

,

d 39, 37, 35,

,

,

f 7, 15, 23,

,

,

,

10 What is 8 squared? 11 What is 3 27 ? 12 Find two numbers that have a product of 48.

3-01 Special number patterns Worksheet 3-02

The numbers 1, 2, 3, 4, 5, … are called the counting numbers. There are groups of counting numbers which make special patterns. We will investigate some of them.

Triangular and square numbers

Exercise 3-01 TLF

L 1936

1 Triangular numbers are shown in the diagram below.

Circus towers: triangular towers

1

78

3

NEW CENTURY MATHS 7

6

10

03 NCM7 2nd ed SB TXT.fm Page 79 Saturday, June 7, 2008 3:51 PM

a Why are they called ‘triangular numbers’? b Work out all the triangular numbers less than 100. c Complete four more lines of this pattern: 1=1 1+2=3 1+2+3=6 1 + 2 + 3 + 4 = 10 d Describe how the pattern in part c works. e Use what you have worked out to help you find the 100th triangular number. (Hint: Do you know a quick way to add up all the numbers from 1 to 100?) 2 Square numbers are shown in the diagram below.

TLF

L 1935

Circus towers: square stacks

1

4

9

16

a Why are these called ‘square numbers’? b Work out all the square numbers up to 100. c Complete four more lines of this pattern: 1=1 1+3=4 1+3+5=9 1 + 3 + 5 + 7 = 16 d Describe how the pattern works. e Work out another pattern to help you find the square numbers. What is the 50th square number? f Complete four more lines of these patterns: i 1 = 12 ii 22 = 12 + (1 + 2) 32 = 22 + (2 + 3) 1 + 2 + 1 = 22 2 42 = 32 + (3 + 4) 1+2+3+2+1=3 g Each square number is said to be the sum of two consecutive triangular numbers. +

3

+

=

6

=

9

Show that this is true for the square numbers up to 100. h Find two numbers that are both triangular numbers and square numbers. CHAPTER 3 EXPLORING NUMBERS

79

03 NCM7 2nd ed SB TXT.fm Page 80 Saturday, June 7, 2008 3:51 PM

Original pair of rabbits

4th generation

2 pairs

Birth

2nd generation

3rd generation

1 pair

Birth

1st generation

1 pair

Birth

Fibonacci numbers

3 Leonardo Fibonacci was an Italian mathematician who lived in the early 13th century. He discovered this pattern when studying the breeding habits of rabbits: 1, 1, 2, 3, 5, 8, 13, 21, . . . The diagram below illustrates this. The vertical arrows labelled ‘Birth’ indicate the new offspring of a pair of rabbits every two months. The unlabelled arrows indicate the same pair of rabbits. After each month, the number of pairs is a term in Fibonacci’s pattern.

Birth

Worksheet 3-03

3 pairs

5 pairs

a How is the Fibonacci pattern formed? b Add five more lines to this pattern: 1 1 1+1=2 1+2=3 2+3=5 3+5=8 c Write the first 20 Fibonacci numbers. i Write every third Fibonacci number, beginning with 2. What number divides evenly into all these numbers? ii Write every fourth Fibonacci number, beginning with 3. What number divides evenly into all these numbers? iii Write every fifth Fibonacci number, beginning with 5. What number divides evenly into all these numbers? d i Find any triangular numbers in the Fibonacci numbers up to 100. ii Find any square numbers in the Fibonacci numbers up to 100. e Pairs of Fibonacci numbers are found by counting along the spirals on pine cones. Investigate how and where else Fibonacci numbers occur in nature.

80

NEW CENTURY MATHS 7

03 NCM7 2nd ed SB TXT.fm Page 81 Saturday, June 7, 2008 3:51 PM

4 Blaise Pascal, a French mathematician who lived 1 in the 17th century, studied a triangle of 1 1 numbers known to the Chinese as the Yanghui 1 2 1 triangle. Each row of the triangle is created using 1 3 3 1 the numbers in the row above it. The triangle is 1 4 6 4 1 known as Pascal’s triangle. The first seven rows 1 5 10 10 5 1 are shown at the right. 6 15 20 15 6 1 a Complete the next four rows of Pascal’s triangle. b Describe how the pattern works. c Add each row in Pascal’s triangle. What do you notice? d The diagonals in Pascal’s triangle produce some interesting patterns. Write the triangular numbers using Pascal’s triangle. Add along the arrows e We can even find Fibonacci 1 to find the Fibonacci numbers in this pattern. Rewrite 1 1 numbers. the triangle above as a right1 2 1 angled triangle. 1 3 3 1 1 1

4 5

6 10

4 10

1 5

1

5 A palindrome is a word, number or sentence that reads the same forward and backward. The following number, words and sentence are all palindromes: noon 151 Able was I ere I saw Elba (Napoleon Bonaparte)

Place names can be palindromes.

a Select the palindromes from these numbers. 447 373 656 281 37 22 899 191 797 b Find the numbers between 1000 and 2000 that are palindromes. c The following steps change any number into a palindrome: • choose any number to start with 64 • reverse the digits and add 46

516

←

• reverse the digits and add

110 011

• repeat until you get a palindrome.

121 CHAPTER 3 EXPLORING NUMBERS

81

Worksheet 3-04 Pascal’s triangle

03 NCM7 2nd ed SB TXT.fm Page 82 Saturday, June 7, 2008 3:51 PM

Find out how many steps it takes to form a palindrome from each of these numbers. i 26 ii 28 iii 47 iv 75 v 149 vi 273 vii 1756 viii 2379 ix 4021 d List some other words and place names that are palindromes.

Using technology

Developing number patterns Follow the instructions below to set up a spreadsheet. • Enter the headings, as shown below, into the given cells in a spreadsheet.

• Enter ‘1’ into cell A2 and ‘2’ into cell A3. Highlight the two cells and Fill Down (click and hold the square in the bottom right-hand corner of cell A3).

• Fill Down to cell A31 (you should see ‘30’ in this cell). Odd numbers 1 Enter the formula =A2 into cell B2. 2 Enter the formula =B2+2 into cell B3. 3 Click on cell B3, and Fill Down to cell B31 to obtain the first 30 odd numbers. Even numbers 1 Enter the formula =A2 into cell C2. 2 Enter the formula =C2+2 into cell C3. 3 Click on cell C3, and Fill Down to cell C31 to obtain the first 30 even numbers. Square numbers 1 Enter the formula =A2^2 into cell D2. 2 Click on cell D3, and Fill Down to cell D31 to obtain the first 30 square numbers. Triangular numbers 1 Enter the formula =A2 into cell E2. 2 Enter the formula =E2+A3 into cell E3. 3 Click on cell E3, and Fill Down to cell E31 to obtain the first 30 triangular numbers.

82

NEW CENTURY MATHS 7

03 NCM7 2nd ed SB TXT.fm Page 83 Saturday, June 7, 2008 3:51 PM

Fibonacci numbers 1 Enter the formula =A2 into cell F2. 2 Enter the formula =F2 into cell F3. 3 Enter the formula =F2+F3 into cell F4. 4 Click on cell F4, and Fill Down to cell B31 to obtain the first 30 Fibonacci numbers. Questions 1 Use your spreadsheet to answer the following questions. a Name the two smallest odd numbers that are also square. Write your answer in cell H1. (Note: separate your answers with a comma.) b Name all the even numbers less than 30 that are also triangular. Write your answer in cell H2. c Find all the triangular numbers less than 60 that are also Fibonacci numbers. Write your answer in cell H3. d Find all the square numbers between 300 and 600. Write your answer in cell H4. e State the 18th Fibonacci number. Write your answer in cell H5. f i In cell H6, write a formula to find the difference between the 24th and 25th Fibonacci numbers. ii What cell in column F corresponds to your answer in (i)? Write your answer in cell H7. g i Extend column A to represent the first 50 numbers. ii Extend the square and triangular numbers to represent the first 50 numbers in each pattern. h Name the first number over 1000 that is both square and triangular. Write your answer in cell H8. Extension: Factorials Factorials: a definition:

1! = 1 2! = 1 × 2 = 2 3! = 1 × 2 × 3 = 6 . . . n! = 1 × 2 × 3 × … × n

1 Enter the formula =A2 into cell G2. 2 Using the definition given, develop a formula for 2! in cell G3, using cells G2 and A3. 3 Click on cell G4, and Fill Down to cell G31 to obtain the first 30 factorial numbers. Displaying large numbers a Highlight all the cells that don’t show full numbers (e.g. 4.79E+08). Right click and choose Format Cells. Change the settings to ‘number’ and ‘0 decimal places’. b ######## in a cell indicates that the column is not wide enough to hold the number with all digits showing. You may need to widen the column until you can see all numbers down to cell G31.

CHAPTER 3 EXPLORING NUMBERS

83

03 NCM7 2nd ed SB TXT.fm Page 84 Saturday, June 7, 2008 3:51 PM

Working mathematically

Reasoning and communicating

Figurate numbers TLF

Numbers formed from geometric shapes, such as triangular or square numbers, are called figurate numbers. There are many figurate number patterns. 1 Investigate the pentagonal numbers.

L 1939

Circus towers: square pyramids

2 Investigate the hexagonal numbers. 3 What are the names of the other types of figurate numbers?

More types of numbers

Worksheet 3-05

Investigate one or more of the following types of numbers and find out the relationships and patterns in them. You may find the Internet useful. Prepare a short talk for the class on your topic. • Amicable numbers • Perfect numbers • The golden ratio/rectangle • Irrational numbers • Pythagorean triads • Binary, octal and hexadecimal numbers • Factorial numbers, for example the meaning of 5!

Perfect and amicable numbers

3-02 Tests for divisibility 8

82

79 is NOT divisible by 3 since 7 + 9 = 16, and 3 does not go evenly into 16.

39 6

04

2 5 8

A number is divisible by 3 if the sum of its digits is divisible by 3.

00 00

78

962

00 0

A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8.

10

Divisibility tests

It is often useful to know if a number is divisible by another number. Here are some simple divisibility tests to help you.

4136

A number is divisible by 5 if it ends in 0 or 5. 26 040

50 005

Worksheet 3-06

279

84

NEW CENTURY MATHS 7

9

3 4 6 7 9 10

A number is divisible by 9 if the sum of its digits is divisible by 9. 171 1+7+1=9 812 754 4 = 27 74 7+5+ 8 592 8 + 1 + 2 +

2 59

13 592 is divisible by 8.

364 805

4506 4 + 5 + 6 = 15

ends in 6

13

A number is divisible by 8 if the last three digits are divisible by 8.

840

A number is divisible by 6 if it is divisible by both 2 and 3. ends in 8 48 { 4 + 8 = 12

67

A number is divisible by 4 if its last two digits are divisible by 4.

32 0

5 4 20

679 320 is divisible by 4.

There is no simple test for divisibility by 7.

90

6 27

A number is divisible by 10 if it ends in 0. 300

20 304050

10

03 NCM7 2nd ed SB TXT.fm Page 85 Saturday, June 7, 2008 3:51 PM

Example 1 Which of the numbers 2 to 10 divide exactly into 112? Solution 2: 112 ends in a 2 so it is divisible by 2. 3: 1 + 1 + 2 = 4: 4 is not divisible by 3, so 112 is NOT divisible by 3. 4: 12 ÷ 4 = 3 so 112 is divisible by 4. 5: 112 does not end in 0 or 5 so is NOT divisible by 5. 6: 112 is not divisible by 3, so it is NOT divisible by 6. 16 7: Check by division: 7 112 so 112 is divisible by 7. 14 8: Check by division: 8 112 so 112 is divisible by 8. 9: 1 + 1 + 2 = 4: 4 is not divisible by 9 so 112 is NOT divisible by 9. 10: 112 does not end in 0 so it is NOT divisible by 10. Answer: 2, 4, 7 and 8 divide exactly into 112.

Exercise 3-02 1 Copy this table and work out which of the numbers from 2 to 10 divide exactly into the given numbers (88 has been done for you). Number

2

3

4

5

6

7

8

9

Ex 1

10

252 600 88

✓

✓

✓

121 6215 3720 747 4753 110 001 40 436 840 75 2 000 646 20 106 7434 601 295

CHAPTER 3 EXPLORING NUMBERS

85

03 NCM7 2nd ed SB TXT.fm Page 86 Saturday, June 7, 2008 3:51 PM

2 Which number is divisible by both 4 and 5? Select A, B, C or D. A 10 B 15 C 20 D 25 3 Write a number less than 100 which is divisible by: a 3 and 5 b 4 and 5 c 6 and 7

d 2 and 6

4 Write a number greater than 100 which is divisible by: a 6 b 5 c 7 d 2 and 3

e 8 and 9

Just for the record

Karl Gauss Karl Gauss was a German mathematician and astronomer who lived from 1777 to 1855. He invented a new way of finding the positions of heavenly bodies and was one of the first to study electricity. Gauss showed his mathematical ability early in life. When he was in primary school, the class was given the task of adding all the numbers from 1 to 100. The teacher thought this would keep the class busy for some time but Gauss was very quick to find the answer and even quicker to explain why he was not working on the problem. This is how he did it: 1 + 2 + 3 + 4 + … + 96 + 97 + 98 + 99 + 100 1 + 100 = 101 2 + 99 = 101 3 + 98 = 101, etc. Find how many pairs of numbers there are and then find the answer to the problem.

Skillsheet 3-02

3-03 Factors

Factors and primes

!

The factors of a number are those whole numbers that divide exactly into it.

Example 2 What are the factors of 12? Solution The possible ways of multiplying to get 12 are: • 4 × 3 or 3 × 4 • 6 × 2 or 2 × 6 The factors of 12 are: 1, 2, 3, 4, 6 and 12. (Note that 1 will be a factor of every number.)

86

NEW CENTURY MATHS 7

• 1 × 12 or 12 × 1

03 NCM7 2nd ed SB TXT.fm Page 87 Saturday, June 7, 2008 3:51 PM

Example 3 Find the highest common factor of 24 and 30. Solution The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 The common factors of 24 and 30 are: 1, 2, 3 and 6. The highest common factor is 6.

The highest common factor (HCF) of two or more numbers is the largest factor that is common to all those numbers.

!

Exercise 3-03 1 In each of these pairs, is the smaller number a factor of the larger number? a 8, 24 b 3, 39 c 4, 42 d 9, 45 e 8, 54 f 7, 91 g 7, 133 h 6, 48 i 5, 57 j 11, 143 2 List all the factors of: a 16 b 21 f 48 g 52 k 28 l 100

Ex 2

c 24 h 80 m 45

d 36 i 112 n 200

e 35 j 144 o 363

3 Which of these numbers is not a factor of 45? Select A, B, C or D. A9 B5 C7 D3 4 Find the common factors for each of these pairs of numbers. a 2, 4 b 9, 6 c 6, 14 e 50, 150 f 46, 69 g 10, 15 i 30, 20 j 18, 24 k 60, 90 m 45, 15 n 36, 39 o 27, 64

d h l p

8, 12 12, 16 39, 26 350, 210

5 Find the common factors for each of these sets of numbers. a 2, 4, 6 b 10, 50, 60 c 22, 33, 121 d 24, 36, 144 e 6, 9, 12 f 16, 24, 40, 56 g 28, 70, 42, 98 h 30, 90, 75, 135 i 50, 60, 90, 120 6 Find the highest common factor for each of these pairs of numbers. a 12 and 60 b 33 and 22 c 132 and 60 d 9 and 21 e 45 and 78 f 64 and 144 g 16 and 12 h 8 and 14 i 50 and 150 j 18 and 24 k 48 and 72 l 15 and 25 m 35 and 21 n 45 and 18 o 75 and 125 CHAPTER 3 EXPLORING NUMBERS

Ex 3

87

03 NCM7 2nd ed SB TXT.fm Page 88 Saturday, June 7, 2008 3:51 PM

7 ‘Every whole number has at least two factors.’ Is this true or false? Why? 8 Using your answers to Question 5, find the highest common factor for each of the given sets of numbers.

Working mathematically

Applying strategies and reasoning

Factor path puzzle 1 Copy this grid, and try to reach the 100 square at the bottom corner, starting at the 200 square in the top corner and using related factors. You can move from one number to another by going sideways, up or down the page (not diagonally), but only if the numbers have a factor (not 1) in common. So you can move from 80 to 65 (common factor 5), but not from 65 to 72 (no common factor).

200

80

65

91

143

156

195

175

32

96

71

110

77

121

35

28

15

209

87

90

21

39

169

117

95

57

37

81

63

11

29

72

76

75

51

14

98

56

132

48

78

85

105

45

44

187

112

221

100

2 Find a factor path starting in the bottom corner (105) and finishing top right (195). 3 Choose different starting and finishing positions. Do they all have connecting factor paths?

Just for the record

Korean mathematics In Korea, school students find the highest common factor (HCF) using the following method. To find the HCF of 24 and 30: divide by the first prime number → divide by the next prime number →

2 24 30 3 12 15 4 5 Since it is not possible to divide any more, stop. The HCF = 2 × 3 = 6 Use this method to find the HCF of each of the following sets of numbers. a 12 and 15 b 18 and 48 c 13, 20 and 28 d 15, 21 and 45 e 8, 12, 16 and 48 f 120 and 250 g 48 and 120 h 96 and 144 i 256, 144 and 48 j 675, 1350 and 825

88

NEW CENTURY MATHS 7

03 NCM7 2nd ed SB TXT.fm Page 89 Saturday, June 7, 2008 3:51 PM

Mental skills 3

Maths without calculators

Calculating differences and making change In every subtraction problem, for example 135 − 47, think of finding the ‘gap’ between the two numbers. That is, find the number in this case that must be added to 47 to get 135. 1 Examine these examples. a 135 − 47 135 47 50 35 Think: 47 + = 135 3 100

50

Count: ‘47, 50, 100, 135’ Add: 3 + 50 + 35 = 88

150

Answer: 135 − 47 = 88 b 244 − 115 Think: 115 +

115 5

= 244

80 150

100

244

44 200

Count: ‘115, 120, 200, 244’ Add: 5 + 80 + 44 = 129 Answer: 244 − 115 = 129 c $60 − $47.65

$47.65 35c $2

$60 $10 $60

$50

$70

Count: ‘$47.65, $48, $50, $60’ Add: $0.35 + $2.00 + $10.00 = $12.35 Answer: $60 − $47.65 = $12.35 d $100 − $88.45

$88.45

55c $1

$80

$90

$10

$100

$100

Count: ‘$88.45, $89, $90, $100 Add: $0.55 + $1.00 + $10.00 = $11.55 Answer: $100 − $88.45 = $11.55 2 Now simplify the following. a 176 − 88 d 425 − 340 g $70 − $58.40 j $100 − $69.95

b e h k

221 − 54 518 − 389 $80 − $73.25 $30 − $22.90

c f i l

670 − 356 199 − 78 $45 − $40.30 $50 − $17.10

CHAPTER 3 EXPLORING NUMBERS

89

03 NCM7 2nd ed SB TXT.fm Page 90 Saturday, June 7, 2008 3:51 PM

Working mathematically

Applying strategies

More factor paths Make your own 7 × 7 factor path grid similar to the one in the ‘Factor path puzzle’ on page 88. Try making a 4 × 4 grid first, then a 5 × 5 grid and then a 7 × 7 grid. Discuss with your teacher the decisions you need to make as you develop your grid.

Skillsheet 3-02

3-04 Prime and composite numbers

Factors and primes

!

A prime number has only two factors: 1 and itself. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, . . . A composite number has more than two factors. The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, . . .

Note: 1 is neither prime nor composite. (It has only one factor.)

Exercise 3-04 Worksheet 3-07 Sieve of Eratosthenes

1 Eratosthenes, a mathematician in ancient Greece, found an easy way to work out prime numbers. It is called the Sieve of Eratosthenes and works by deleting multiples of numbers. (Use the link to go to a spreadsheet version of the Sieve.) a Copy the grid below or print out Worksheet 3-07. 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100 101 102 103 104 105 106 107 108

109 110 111 112 113 114 115 116 117 118 119 120

b Cross out 1. It is neither prime nor composite. c Except for 2, colour all the multiples of 2 red.

90

NEW CENTURY MATHS 7

03 NCM7 2nd ed SB TXT.fm Page 91 Saturday, June 7, 2008 3:51 PM

d Except for 3, colour all the multiples of 3 green. e Continue, with different colours, until there are no more multiples. What do you notice about the numbers that are not coloured? 2 Divide these numbers into two groups (primes and composites).

2 47

33

04

5 67

71

9

59

62

27

41

9

99

1 129 17

10

2064

77

10

3 Write any whole numbers which are neither prime nor composite. 4 a b c d

List the prime numbers between 36 and 50. List the composite numbers between 65 and 80. List the prime numbers less than 20. List the composite numbers larger than 30 but less than 47.

5 Which number is divisible only by prime numbers, itself and 1? Select A, B, C or D. A 12 B 14 C 16 D 18 6 Look up other meanings for the word ‘composite’. Suggest why this word is used the way it is in mathematics.

3-05 Prime factors Every composite number can be written as a product of its prime factors. The prime factors can be found by using a factor tree.

Worksheet 3-08 Factor trees

Example 4 Skillsheet 3-03

Write 24 as a product of its prime factors. Solution Factor tree

3

Prime factors by repeated division

24 3

×

8

3

×

2

×

4

×

2

×

2

×

3 and 8 are factors 2 and 4 are factors of 8 3 is prime 2

2 is a factor of 4 2 is prime—stop

As a product of prime factors, 24 = 2 × 2 × 2 × 3.

CHAPTER 3 EXPLORING NUMBERS

91

03 NCM7 2nd ed SB TXT.fm Page 92 Saturday, June 7, 2008 3:51 PM

Example 5 Find the highest common factor (HCF) of 1960 and 2000. Solution 1960 ×

10

2000 196

2

2 × 5 × 4 × 49 2 ×

2

5 × 2 × 2 × 7 × 7

2

1960 = 2 × 2 × 2 × 5 × 7 × 7

× 100 10

×

1000 ×

10

10

×

10

× 5 × 2 × 5 × 2 × 5 × 2

2 Both numbers contain 2 × 2 × 2 × 5. The HCF is 2 × 2 × 2 × 5 = 40.

×

×

2000 = 2 × 2 × 2 × 2 × 5 × 5 × 5

Example 6 Write 648 as a product of its prime factors, using index notation (powers). Solution 648

2 2

So

2

×

324

×

4

×

81

648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 23 × 34

34 is read ‘3 to the power 4’ and 4 is called the power or index.

× 2 × 2 × 9 × 9

2 × 2 × 2 × 3 × 3 × 3 × 3

Exercise 3-05 Ex 4

1 Use factor trees to express each of these numbers as a product of its prime factors. a 8 b 63 c 45 d 36 e 51 f 49 g 90 h 27 i 130 j 200 k 275 l 342 m 1250 n 1020 o 837

92

NEW CENTURY MATHS 7

03 NCM7 2nd ed SB TXT.fm Page 93 Saturday, June 7, 2008 3:51 PM

2 What are the prime factors of 1260? Select A, B, C or D. A 2×3×3×3×5×7 B 2×2×2×3×5×7 C2×2×3×3×3×7 D2×2×3×3×5×7 3 Find the highest common factor of each of these pairs of numbers. a 324 and 486 b 6000 and 1260 c 2475 and 3375 d 4900 and 1960 e 4950 and 1530 f 1404 and 900

Ex 5

4 Use factor trees to write each number as a product of its prime factors in index notation. a 18 b 20 c 45 d 72 e 98 f 196 g 32 h 135 i 200 j 900

Ex 6

Just for the record

Big numbers Here are the names of some very large numbers. Name

Numeral

Power of 10

one

1

100

ten

10

101

hundred

100

102

thousand

1000

103

million

1 000 000

106

billion

1 000 000 000

109

trillion

1 000 000 000 000

1012

1 000 000 000 000 000

1015

1 000 000 000 000 000 000

1018

sextillion

1 000 000 000 000 000 000 000

1021

septillion

1 000 000 000 000 000 000 000 000

1024

1 000 000 000 000 000 000 000 000 000

1027

nonillion

1 000 000 000 000 000 000 000 000 000 000

1030

decillion

1 000 000 000 000 000 000 000 000 000 000 000

1033

quadrillion quintillion

octillion

Find the names of some numbers greater than a decillion.

CHAPTER 3 EXPLORING NUMBERS

93

03 NCM7 2nd ed SB TXT.fm Page 94 Saturday, June 7, 2008 3:51 PM

Working mathematically

Applying strategies and reasoning

Goldbach’s conjecture In 1742, Christian Goldbach said: Every even number greater than 2 can be written as the sum of two prime numbers. Show that his theory is true for all the even numbers between 1 and 100. Primes may be repeated, for example 10 = 5 + 5, and a number can have more than one pair of prime numbers.

Skillsheet 3-04 Square roots and cube roots

Worksheet 3-09

3-06 Squares, cubes and roots Finding square roots and cubes Raising a number to the power 2 gives its square. Raising a number to the power 3 gives its cube.

Powers and roots

Example 7 Find a the square of 11

b the cube of 5

Solution a 112 = 11 × 11 = 121 The square of 11 is 121.

b 53 = 5 × 5 × 5 = 125 The cube of 5 is 125.

The x 2 , x 3 and ^ keys on a calculator can be used to find the square, cube and other powers of a number.

Finding square roots and cube roots The process that undoes squaring is finding the square root (symbol ). The square root of a given number is the positive value which, when multiplied by itself, produces the given number. The process that undoes cubing is finding the cube root (symbol 3

Square root sign first used in 1220

94

NEW CENTURY MATHS 7

).

Cube root sign created in 1525

03 NCM7 2nd ed SB TXT.fm Page 95 Saturday, June 7, 2008 3:51 PM

Example 8 What is: a the square root of 64?

b the cube root of 27?

Solution a The square root of 64 =

64 = 8 (because 8 × 8 = 64).

b The cube root of 27 = 3 27 = 3 (because 3 × 3 × 3 = 27).

Example 9 Estimate the value of

40.

Solution There is no exact answer for the square root of 40, because there isn’t a number which, if squared, equals 40 exactly. Instead, we find a number whose square is close to 40. Looking at the square numbers 52 = 25, 62 = 36, 72 = 49, we can tell 40 must be between 6 and 7. Because 40 is closer to 36 than to 49, the square root must be closer to 6. 40 ≈ 6.3.

As an estimate,

The

and

3

keys on a calculator can be used to find the square root and cube 40 , press

root of a number. To calculate

= . The result is 6.324555…,

40

a more accurate answer than our estimate above.

Finding square roots and cube roots using a factor tree

Example 10 Use a factor tree to find the value of

196 .

Solution

4

×

196 = 2 × 2 × 7 × 7

So

196

196 = 2 × 2 × 7 × 7 =2×7 = 14

49

2 × 2 × 7 ×

7

(Note:

2 × 2 = 2)

CHAPTER 3 EXPLORING NUMBERS

95

03 NCM7 2nd ed SB TXT.fm Page 96 Saturday, June 7, 2008 3:51 PM

Example 11 Use a factor tree to find the value of 3 216 . Solution 216 4 2 × 2 2 ×

216 = 2 × 2 × 2 × 3 × 3 × 3

So 3

×

54

× 2 × 27

2 × 2 × 3

2 × 2 × 2 ×

216 = 3 2 × 2 × 2 × 3 × 3 × 3 =2×3 =6

(Note: 3 2 × 2 × 2 = 2)

× 9

3 × 3 × 3

Exercise 3-06 Ex 7

1 Copy and complete the following table. Number

1

2

3

4

5

6

7

8

9

10

11

16

Number squared

512

Number cubed

2 Which number(s) from Question 1 are both square and cube numbers? 3 Use your calculator to find the square of each of these numbers. a 84 b 123 c 24 d 42 4 Use your calculator to find: a 112 b 153 d 673 e 0.12 Ex 8

5 Find the square root of: a 9 b 16 e 25 f 4

c 1002 f 3.53 c 81 g 36

d 121 h 100

Ex 8

6 Find the cube root of each of these numbers, using the table from Question 1. a 8 b 125 c 343 d 1000 e 729 f 27 000

Ex 9

7 Between which two numbers does calculator). A 4 and 5

B 5 and 6

8 Between which two numbers does A 14 and 16 B 4 and 5

96

NEW CENTURY MATHS 7

27 lie? Select A, B, C or D (without using a C 26 and 28

D 756 and 784

15 lie? Select A, B, C or D. C 196 and 225 D 3 and 4

12

03 NCM7 2nd ed SB TXT.fm Page 97 Saturday, June 7, 2008 3:51 PM

9 Between which two consecutive whole numbers does

80 lie?

10 Give estimates for the following, then use a calculator to check. a

50

d 3 66

b

142

c

e 3 999

1000

f 3 123

11 Find the following square roots, using factor trees.

Ex 10

a

484

b

1764

c

625

d

900

e

784

f

256

g

196

h

400

i

3136

12 Find the following, using a factor tree. a

3

10 648

d 3 64 000

b

3

Ex 11

2744

c

e 3 4913

3

3375

f 3 9261

Using technology

Absolute cell referencing Using a spreadsheet to calculate powers 1 Set up your spreadsheet by entering the information in the cells as shown below.

2 We need to use absolute cell referencing to complete this task easily. We use this technique to maintain a particular value in a cell without changing it when writing a formula. For example: to write 21 in cell B2, enter =$B$1^A2. This formula will not change the 2, but will change the power to each consecutive number as we Fill Down (i.e. 21, 22, 23, etc.) 3 Click on cell B2 and Fill Down to cell B13. Your spreadsheet will now show the first 12 powers of 2. 4 By modifying the formula given in point 2, repeat this process, using the appropriate cells, absolute cell referencing and Fill Down for columns C, D and E to show the first 12 powers of 3, 5 and 7.

CHAPTER 3 EXPLORING NUMBERS

97

03 NCM7 2nd ed SB TXT.fm Page 98 Saturday, June 7, 2008 3:51 PM

Displaying large numbers If the numbers cannot be seen properly follow these steps: a Highlight all the cells that don’t show full numbers (e.g. 2·44E + 08). Right click and choose Format cells. Change the settings to ‘number’ and ‘0 decimal places’. b ######## in a cell indicates that the column is not wide enough to hold the number with all digits showing. You may need to widen the column until you can see all numbers.

Using a spreadsheet to calculate the lowest common multiple 5 This activity finds the lowest common multiple (LCM) of sets of numbers. The LCM is the smallest number that all numbers in a particular set divide into; e.g. for the pair of numbers 6 and 20, the LCM = 60. This is the smallest number both 6 and 20 divide into. Find the LCM of 8, 12 and 16. Open a new sheet and enter the information shown.

a In cell B2, enter the formula =$B$1*A2. Use Fill Down to find the first 15 multiples of 8. b In cells C2 and D2, enter similar formulas and Fill Down to find the first 15 multiples of 12 and 16. [Hint: Only change the absolute cell reference.] c Now, compare the columns and identify the LCM of 8, 12 and 16. 6 Modify your spreadsheet from part a above to find the LCM of the following sets of numbers. Note: you may need to extend beyond the first 15 multiples. a 6 and 15 b 12 and 18 c 3, 7 and 15 Try other combinations of numbers and calculate each LCM.

d 48, 60 and 75

Power plus 1 Evaluate each of the following. b 8 × 42 a 72 − 33 d 33 + 62 − 42 e 34 + 33 + 32 + 3

98

NEW CENTURY MATHS 7

c 53 × 22 f 12 + 22 + 32 + 42

03 NCM7 2nd ed SB TXT.fm Page 99 Saturday, June 7, 2008 3:51 PM

2 Arrange each of these sets of index terms in order, from the smallest value to the largest. a 23, 32, 35, 53, 25, 52 b 44, 73, 35, 82, 52, 63 c 1002, 114, 27, 34, 54 3 a Copy and complete: 12 = 112 = 1112 = 11112 = b Based on the patterns in your part a answers, write the squares of these numbers: i 11 111 ii 1 111 111 iii 111 111 111 iv 1 111.1111 4 a Copy and complete this number pattern. 1 = 1 = (1)3 + (2)3 2+3+4 = 1+ 5 + 6 + 7 + 8 + 9 = 8 + 27 = ( )3 + ( )3 10 + 11 + 12 + 13 + 14 + 15 + 16 = + = ( )3 + ( )3 b Write the next two lines of the pattern in part a. c Find the sum without adding each time. Show how you did it. ii 82 + 83 + … + 99 + 100 i 50 + 51 + … + 63 + 64 iv 577 + 578 + … + 624 + 625 iii 290 + 291 + … + 323 + 324 5 Try finding the square root of each number. (They’re not as hard as they look!) a 2500 b 8100 c 10 000 d 1 000 000 e 1 210 000 f 100 000 000 g 640 000 h 176 400 i 10 000 000 000 6 Find the cube root of each of these numbers. (What you discovered in Question 5 should help.) a 8000 b 343 000 c 1 000 000 d 64 000 000 e 1 000 000 000 f 27 000 7 Find the square root of: a 3×3×2×2 c 36 × 49

b 5×5×4×4×3×3 d 16 × 25 × 4

8 Find the cube root of: a 2×2×2 c 7×7×7×6×6×6 e 125 × 64 × 1000

b 4×4×4×5×5×5 d 8 × 27 f 343 × 729

9 Find the value of: a

92

d 3 29

b

34

e 3 33 × 56

c

54 × 26

f 4 16

10 Two prime numbers that differ by 2 are called twin primes. For example, 11 and 13 are twin primes, but 23 and 29 are not. Find the sets of twin primes between 1 and 100.

CHAPTER 3 EXPLORING NUMBERS

99

03 NCM7 2nd ed SB TXT.fm Page 100 Saturday, June 7, 2008 3:51 PM

Chapter 3 review Worksheet 3-10

Language of maths composite number estimate highest common factor power square

Exploring numbers crossword

cube factor index notation prime factors square root

cube root factor tree palindrome prime number triangular number

divisibility test Fibonacci number Pascal’s triangle product

1 Describe in your own words how the Fibonacci numbers are formed. 2 Find the non-mathematical meaning of: a ‘factor’

b ‘index’

3 The date 30/11/03 was a palindromic date. When will be the next palindromic date? 4 Describe what a ‘factor tree’ does. 5 Find as many meanings for these words as you can. a product b prime

Topic overview • Write in your own words what you have learnt about number patterns and the way numbers behave. • What was your favourite part of this topic? • What parts of this topic did you not understand? Talk to your teacher or a friend about them. • Give examples of where some of the number patterns in this chapter occur or are used. • This diagram provides a summary of this chapter of work. Copy it into your workbook and complete it. Use bright colours, add your own pictures, and change it, if necessary, to be sure you understand it. Triangular Number patterns

Fa

cto

rs

Fibonacci

is

Div

Exploring numbers Sq

trees

ua

m po Co

e

NEW CENTURY MATHS 7

im

100

Pr

sit e

r Facto

sts

y te

it ibil

res

,c

ub

es,

ro

ots

03 NCM7 2nd ed SB TXT.fm Page 101 Saturday, June 7, 2008 3:51 PM

Chapter revision

Topic test 3

1 Write the next three numbers in each of these patterns. a 1, 3, 5, 7, … b 2, 4, 6, 8, … d 1, 4, 9, 16, … e 1, 1, 2, 3, 5, 8, … 2 In Question 1, which set of numbers are the: a square numbers? b triangular numbers? 3 Write the following. a a triangular number between 10 and 20 c the next palindrome after 2002 e the first five composite numbers

Exercise 3-01

c 1, 3, 6, 10, … f 60, 55, 50, 45, … Exercise 3-01

c Fibonacci numbers? Exercise 3-01

b the highest Fibonacci number below 40 d the next prime number after 29 f the square number between 40 and 50

4 Write the next three lines of Pascal’s triangle as shown on the right.

1

1

1

1 1 1 5 Which of the numbers from 2 to 10 divide exactly into: a 81? b 327? c 228? d 170?

2 3

Exercise 3-01

1 3

4

6

1 4

1 Exercise 3-02

e 4326?

6 a Write all the factors of 60. b Write all the factors of 42.

Exercise 3-03

7 a Find the common factors of 42 and 60. b Find the highest common factor of 20 and 48. c Find the highest common factor of 36 and 84.

Exercise 3-03

8 Find the prime numbers from: 27 93 6 29 19 100 65 37 17 13

Exercise 3-04

39 1

96 67

31 73

57 83

9 Draw factor trees to find the prime factors of these numbers. a 24 b 60 c 27 e 36 f 45 g 72

2 89

51 27 Exercise 3-05

d 200 h 144

10 Write your answers from Question 9 using index notation. 11 Find the square of each of the following. a 4 b 9

Exercise 3-06

c 6

d 11

12 Find the square root of each of the following. a 64 b 25 c 49 13 Find the cube of each of the following. a 2 b 6

Exercise 3-06

d 144 Exercise 3-06

c 9

d 8

14 Find the cube root of each of the following. a 27 b 64 c 125 15 Use factor trees to find: a 225 b

Exercise 3-05

Exercise 3-06

d 1000 Exercise 3-06

256

c

1764

16 Between which two consecutive whole numbers does

d

3

5832

55 lie?

CHAPTER 3 EXPLORING NUMBERS

Exercise 3-06

101

03 NCM7 2nd ed SB TXT.fm Page 102 Saturday, June 7, 2008 3:51 PM

Mixed revision 1 Exercise 1-03

1 Use our Hindu–Arabic numerals to write the Babylonian number on the right.

Exercise 1-04

2 Write the Roman numeral XXIX using Hindu–Arabic numerals.

Exercise 1-05

3 Use Hindu–Arabic numerals to write the Chinese number shown on the right.

Exercise 1-06

4 What is the value of the digit 6 in each of the following? a 261 b 1006 c 63 110

Exercise 1-07

Exercise 1-08

5 Write each of the following in expanded notation. a 38 b 201 6 Complete these number grids. a 12 + 3

d 210 632 c 3987

b top row − side column −

9

17

9

15

27 11

c

× 7

d top row ÷ side column

9 35

÷

28

4

27

36 12 6

40 Exercise 1-09

Exercise 1-10

Exercise 1-11

7 Find the answers to each of these. a 390 ÷ 15 b 294 ÷ 21

c 259 ÷ 14

8 Evaluate each of the following. a 15 − 3 × 5 b 7×3+2×5 d 7 + 5 × (12 − 3) e 414 ÷ 18

c 26 ÷ 2 − 14 ÷ 7 f [(3 + 5) × 2 − (20 ÷ 5)] × 5

9 True or false? a 12  20

b 100  25 × 5

c

e 12 × 5  4 × 15

f 3 3 8

d Exercise 2-02

62

−6=6

10 Measure these angles. a

102

NEW CENTURY MATHS 7

b

100 = 10

c

03 NCM7 2nd ed SB TXT.fm Page 103 Saturday, June 7, 2008 3:51 PM

11 Draw an example of: a an obtuse angle

Exercise 2-04

b an acute angle

c a reflex angle

12 Name this angle using three letters and write the name of its parts.

P

Exercise 2-01

b

13 Find the complement of: a 66° b 85° c 12° d 89°

Exercise 2-05

a b

S

14 Find the supplement of: a 32° b 90°

A Exercise 2-05

c 105°

d 153°

15 Draw a diagram and mark in vertically opposite angles.

Exercise 2-06

16 a b c d

Exercise 2-11

Draw a pair of parallel lines and cut them with a transversal. Mark a pair of alternate angles with (•). Mark a pair of corresponding angles with (*). Mark a pair of co-interior angles with (+).

17 Find the size of each angle marked by a letter. a b c

Exercise 2-12

d

b°

50°

p° m°

98° 142°

110° a°

e

f p°

g

a°

h m°

88°

36°

38°

70° c°

m°

b°

i

p°

j x°

m° 56°

k

l 58°

64° x° 75° 45°

37°

y°

18 Write the next three numbers in each of these patterns. a 1, 4, 7, … b 1, 3, 6, … c 1, 1, 2, 3, 5, … 19 Which of the numbers from 2 to 10 divide exactly into: a 68? b 294? 20 Find the factors of: a 18

122°

Exercise 3-01

d 11, 9, 7, … Exercise 3-02

c 6152? Exercise 3-03

b 45

c 360

21 Using factor trees, write each of the numbers in Question 20 as a product of its prime factors. Write your answers using index notation.

Exercise 3-05

22 Simplify each of the following. a 62 b 33 c

Exercise 3-06

25

d

121

e

3

125

f

3

64

MIXED REVISION 1

103