Download Introduction to Number Patterns

Number patterns
Introduction to number patterns
Special groups of whole numbers
7
Fibonacci sequence
9
Factors and multiples
10
The question of divisibility
12
Suggested answers to activities
4930CD: 3 Recognising Everyday Mathematics
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Introduction to number patterns
Everyday tiling patterns are made by interesting repetitions of basic shapes.
Draw what comes next in this tiling pattern.
4930CD: 3 Recognising Everyday Mathematics
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Numbers can make some interesting patterns too:
1
Try these, using your calculator:
37  3 = ___
37  6 = ___
37  9 = ___
37  12 = ___
Keep the pattern going:
37  ___ = ___
___  18
= ___
___  ___ = ___
37  24
2
= ___
How about this one? Using your calculator, write down the answer to these:
1  9 = ___
2  9 = ___
3  9 = ___
4  9 = ___
Keep it going:
________________
________________
________________
________________
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Now for something a little more adventurous:
1  9 + 2 = 11
12  9 + 3 = ___
123  9 + 4 = ___
1234  9 + 5 = ___
Keep going:
________________
________________
1
37  3 = 111
37  6 = 222
37  9 = 333
37  12 = 444
Keep the pattern going:
37  15 = 555
37  18 = 666
37  21 = 777
37  24 = 888
2
1  9 = 0.1111…
2  9 = 0.2222…
3  9 = 0.3333…
4  9 = 0.4444…
Keep it going:
5  9 = 0.5555…
6  9 = 0.6666…
7  9 = 0.7777…
8  9 = 0.8888…
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1  9 + 2 = 11
3
12  9 + 3 = 111
123  9 + 4 = 1111
1234  9 + 5 = 11 111
12 345  9 + 6 = 111 111
123 456  9 + 7 = 1 111 111
Completing a pattern is as easy as identifying similarities and
differences.
For example, write the next two numbers for the number pattern.
1, 3, 9, 27, ____ , ____
Each term is three times the term before it. That is,
Thus the pattern is:
1, 3, 9, 27, 81, 243.
Activity 3A
Write down the next two numbers in each of the following number patterns.
1
5, 15, 25, 35,
2
3, 6, 12, 24,
3
1, 3, 5, 7,
4
10, 7, 4, 1,
5
101, 201, 301,
6
96, 48, 24,
7
80, 70, 61, 53, 46,
8
4, 11, 18, 25,
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1, 4, 9, 16, 25,
10 1, 3, 6, 10.
Check your answers against those at the end of the topic.
Special groups of whole numbers
You might have recognised some of the patterns in Activity 3A as belonging
to special groups of whole numbers.
Odd numbers
The pattern 1, 3, 5, 7, … for example, is the group known as odd numbers.
If we were to draw a diagram representing the group of odd numbers, it
would look like this:
 there is always one
‘unpaired’ dot
Even numbers
The other half of the group of counting numbers is called the group of even
numbers. A diagram of the first few even numbers looks like this:
 all dots are ‘paired’
Square numbers
Did you recognize the pattern 1, 4, 9, 16, 25, …?
This is the group of square numbers, found by multiplying each of the
counting numbers by itself.
4930CD: 3 Recognising Everyday Mathematics
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1
What is the 7th square number?
________________________
________________________
2
What is the 12th square number?
________________________
________________________
1
72 = 7  7 = 49
2
122 = 12  12 = 144
It is easy to see where the name ‘square’ comes from, if you look at the
pattern square numbers make.
Triangular numbers
Did you recognize the pattern 1, 3, 6, 10, …?
You should have found that the pattern is formed by adding one more each
time: That is, add 2, then add 3, then add 4, then add 5, and so on.
Mathematicians love their shapes so much that they call this group
triangular numbers, but they should be called ‘ten-pin bowling numbers’,
don’t you think?
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Fibonacci sequence
This pattern is more challenging. See if you can write down the next two numbers:
1, 1, 2, 3, 5, 8, 13, 21, ____ , ____, …
Can you write the rule, in words?
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
The next two numbers are 34, 55.
Except for the first two in the pattern, each term is the sum of the two terms
before it.
That is
1+1=2
1+2=3
2+3=5
3+5=8
5 = 8 = 13
8 + 13 = 21 and so on.
This pattern of numbers is called the Fibonacci sequence.
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Activity 3B
1
Complete this table for the special groups of numbers above:
Number group
Pattern
(a) even numbers
2, 4, 6, 8, ….
(b) odd numbers
(c) square numbers
(d) triangular numbers
2
(a) What is the 20th square number?
(b) What is the 8th triangular number?
3
34, 25, 43, 100, 81, 27,
111, 2, 70, 9, 64, 49
Which of the numbers in the box above are:
(a) even numbers?
(b) odd numbers?
(c) square numbers?
(d) both square and even?
Check your answers against those at the end of the topic.
Factors and multiples
A factor of a counting number divides into it exactly, that is, with no
remainder. For example, 3 is a factor of 15 because it divides into 15 exactly
5 times while 4 is not a factor of 15 because 15  4 = 3 and something left
over.
Some numbers have several factors, others do not.
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Example 1
List the factors of 15.
Answer:
1  15 = 15 and 3  5 = 15.
So the factors are 1, 3, 5, 15
Example 2
List the factors of 36.
Answer: 1  36 =36
2  18 = 36
3  12 = 36
4  9 = 36
6  6 = 36.
So the factors of 36 are
1, 2, 3, 4, 6, 9, 12, 18, 36.
Example 3
List the factors of 17.
Answer: 1  17 = 17.
So there are only 2 factors of 17: 1 and 17.
Some comments

A number like 17 with only 2 factors (itself and 1) is called a prime
number.

Numbers like 15 and 36, which have more than 2 factors, are called
composite numbers.
List all the prime numbers less than 20.
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2, 3, 5, 7, 11, 13, 17, 19.
Did you wonder what to do with the number 1? 1 is neither prime nor
composite.
A multiple of a counting number is found when you multiply that counting
number by any other counting number.
For example, 10, 20, 30, 40, 50, …………… are all multiples of 10.
3, 6, 9, 12, 15, …………… are all multiples of 3.
List the multiples of 7 between 20 and 29.
__________________________________
21, 28
The question of divisibility
Determining whether or not one number is a factor of another number is
really just a matter of knowing your times tables! But there are some other
helpful tricks, too. Here are some of them.

Do you remember the group of even numbers we looked at earlier? It
was the group that looked like this.
2, 4, 6, 8, 10, …
All even numbers are divisible by 2.
2 is a factor of any number that ends in 2, 4, 6, 8, or 0.
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
Here is an interesting fact. A number is divisible by 3 if its digits add up
to a number divisible by 3. For example, is 141 divisible by 3? Test:
1 + 4 + 1 = 6, and 6 is divisible by 3, so yes, 141 is divisible by 3.
3 is a factor of any number whose digits add to be a multiple of 3.

No matter how large a number you are testing, the number will be
divisible by 4 if its last 2 digits are divisible by 4.
What a time saver! For example, does 4 divide evenly into 13 580?
Test the last 2 digits. 4 does divide evenly into 80, so 4 must also divide
evenly into 13 580.
4 is a factor of any number whose last 2 digits make a multiple of 4.

The group of multiples of 5 is always easy to identify.
5, 10, 15, 20, 25, 30, 35, …
They all end in a ‘5’ or a ‘0’.
5 is a factor of any number which ends in 0 or 5.

Multiples of 10 are even easier to see.
10, 20, 30, 40, 50, …
They all end in ‘0’.
10 is a factor of any number which ends in 0.
Challenge
You probably know how to spot multiples of 11 that are under 100: 11, 22,
33, 44, 55, …They are all double-digit numbers.
Now, with this nifty party trick, you might be able to identify some in the
hundreds as well.
Try this short-cut way to multiply a two-digit number by 11. Take as an
example, the number 23.
Write 2 on the left, leave a space, then write 3 on the right:
2_3
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Now add the 2 and the 3 to get 5, and write the 5 in the gap you have left in
the middle:
253
Result: 23  11 = 253
Check on your calculator.
Try another:
52  11 = ___________
And another:
81  11 = __________
Were your answers 572 and 891?
Another challenge
What happens to our rule when the sum of the two digits is 10 or more?
This means you will have to carry.
Example
47  11 = ?
Use your calculator to find the answer, then see if you can find the rule.
Test your theory on several more examples. Write your test examples and
your rule in the space below:
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_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
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Activity 3C
1
Complete this table by giving the meaning of each word.
Word
Meaning
factor
multiple
prime
composite
divisible
2
Use your definitions in Question 1 to help answer these questions.
(a) List the factors of 24.
(b) Write the first 5 multiples of 4.
(c) Write the multiples of 3 between 20 and 30.
(d) Write down the prime numbers between 20 and 30.
(e) From this list, circle the composite numbers: 20 23 25 27 30 31 33 35 37.
(f) Circle true or false for each of the following statements.
(i)
4 is a factor of 348.
T
F
(ii)
1255 is divisible by 3.
T
F
(iii) 1 000 000 is a multiple of 5.
T
F
(iv) 27 457 is an even number.
T
F
(g) List the factors of 49.
3
There is one even number that is not composite. What is it?
4930CD: 3 Recognising Everyday Mathematics
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4
(a) Complete the table:
Factors of 20
Factors of 32
(b) For the numbers 20 and 32, common factors are factors common to both lists.
(i) List the common factors of 20 and 32.
The highest common factor is the largest number found on both lists.
(ii) What is the highest common factor of 20 and 32?
5
(a) Complete the table for multiples less than 80:
Multiples of 10
Multiples of 12
(b) The lowest common multiple is the smallest number common to both lists.
What is the lowest common multiple of 10 and 12?
6
Lengths of framing timber are sold in multiples of 30 cm. Emma needs 80 cm. What is
the shortest length she can buy for the job?
7
The Olympic Games are held in leap years, which occur every 4 years.
Sydney held the Games in 2000.
(a) Will 2050 be a leap year?
(b) What is the first leap year after 2025?
(c) What will be the year of the first Olympic Games held after 2110?
Check your answers against those at the end of the topic.
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Suggested answers to activities
Activity 3A
1
45, 55
2
48, 96
3
9, 11
4
–2, –5
5
401, 501
6
12, 6
7
40, 35
8
32, 39
9
36, 49
10 15, 21
Activity 3B
1
(b) 1, 3, 5, 7, 9, 11, 
(c) 1, 4, 9, 16, 25, 36, 
(d) 1, 3, 6, 10, 15, 21, 
2
(a) 20 × 20 = 400
(b) 36
3
(a) 34, 100, 2, 70, 64
(b) 25, 43, 81, 27, 111, 9, 49
(c) 25, 100, 81, 64, 49, 9
(d) 100, 64
4930CD: 3 Recognising Everyday Mathematics
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Activity 3C
1
A factor of a counting number divides into it exactly.
A multiple of a counting number is found when you multiply that
number by another counting number.
A prime number has only two factors (itself and 1).
A composite number has more than two factors.
Divisible: A number that can be divided exactly by another number
is said to be divisible by that number.
2
(a) 1, 2, 3, 4, 6, 8, 12, 24
(b) 4, 8, 12, 16, 20
(c) 21, 24, 27
(Note: Although 30 is a multiple of 3, it is not included because the
word ‘between’ excludes the end points.)
(d) 23, 29
(e) 20, 25, 27, 30, 33, 35
(f) (i)
True
(ii)
False
(iii) True
(iv) False
(g) 1, 7, 49
3
2
4
(a)
(b)
Factors of 20
1, 2, 4, 5, 10, 20
Factors of 32
1, 2, 4, 8, 16, 32
(i) 1, 2, 4
(ii) Highest common factor is 4.
5
(a)
Multiples of 10
10, 20, 30, 40, 50, 60, 70
Multiples of 12
12, 24, 36, 48, 60, 72
(b) Lowest common multiple is 60.
6
90 cm
7
(a) No (2050 is not divisible by 4.)
(b) 2028 (Make the last two digits a multiple of four.)
(c) 2112 (Make the last two digits a multiple of four.)
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4930CD: 3 Recognising Everyday Mathematics
 DET NSW, 2005/008/012/06/2006
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