Download the free PDF resource

Complete this pattern to create the first 5
square numbers:
Revising sequences
1
4
9
16
25
List the remaining square numbers up to 100.
1
4
9
16 25 ...
36 49 64 81 100
© www.teachitmaths.co.uk 2011
1
16368
3
6
10
2
15
3
6
3
5
7
11 13 ...
17
19
23
29
31
37
41
47
53
59
61
67
71
73
43
What is the name given to this sequence?
Prime numbers
List the next five triangular numbers.
1
2
16368
Continue this sequence for another 15 terms:
Complete this pattern to create the first 5
triangular numbers:
1
© www.teachitmaths.co.uk 2011
Is there a pattern enabling you to predict the next
number?
No!
10 15 ...
21 28 36 45 55
If printed in font size 12, the largest known prime number would
stretch more than 10 miles!
© www.teachitmaths.co.uk 2011
3
16368
This sequence is called the Fibonacci sequence:
1
1
2
3
5
8 ...
© www.teachitmaths.co.uk 2011
1
1
55
89
1÷1=
Add the previous two terms.
2÷1=
Find the next 10 Fibonacci numbers.
34
3÷2=
55
89
5÷3=
144 233 377 610 987
© www.teachitmaths.co.uk 2011
16368
2
3
144
5
4
8
233
13
377
21
34
610
987
Use the Fibonacci sequence to create a new sequence:
How do you find the next term in the sequence?
... 13 21
16368
8÷5=
5
© www.teachitmaths.co.uk 2011
What do you notice?
The new sequence converges on
one number, 1.618 ..., called the
golden number. It is a ratio that
appears everywhere in the
natural world, from flowers to
snail shells.
16368
6
Predict the next row of numbers in this triangle of
numbers.
1
How does the triangle work?
Patterns in Pascal’s triangle
1
1
1
1
1
1
© www.teachitmaths.co.uk 2011
7
16368
Pascal’s triangle
Pascal did not discover this
triangle, but he was
renowned for studying it.
Blaise Pascal
19th June 1623 – 19th August 1662
1
 In Italy, it is called
Tartaglia’s triangle.
 In China, it is called
Yang Hui’s triangle.
1
1
1
1
1
1
1
1
5
7
15
21
28
© www.teachitmaths.co.uk 2011
6
10
1
4
10
20
35
56
36
35
84
35
7
1
1
1
16368
1
9
3
Add consecutive
triangular numbers.
1
1
5
7
10
21
28
© www.teachitmaths.co.uk 2011
6
15
1
1
1
1
1
© www.teachitmaths.co.uk 2011
6
7
8
6
15
21
28
16368
4
20
1
5
15
35
70
10
35
1
5
15
35
70
1
6
21
56
1
7
28
1
8
1
16368
10
Pascal’s triangle
1
6
1
1
1+1= 2
1
7
28
1
1
1
21
56
1
1
1
10
35
56
1
3
10
1
1
2
4
5
9
1
4
20
56
Add along the
shallow diagonals.
1
3
36
1
1
3
4
6
8
2
1
1
1
8
1
1
Where is the sequence of
Fibonacci numbers hiding in
the triangle?
Pascal’s triangle
28
8
1
1
8
56
1
7
126 126 84
1
1
1
28
21
1
1
21
56
35
1
6
Pascal’s triangle
1
6
1
5
15
70
What is the sequence of
numbers in red?
Triangular numbers
5
10
20
56
1
4
16368
1
15
70
Where is the sequence of
square numbers hiding in the
triangle?
9
28
© www.teachitmaths.co.uk 2011
3
10
21
1
6
15
What is the sequence of
numbers in green?
1
3
4
6
8
2
3
6
8
3
4
5
7
2
Positive integers
1
In other countries, the
triangle is named after
different mathematicians:
1
1
Each number is the
sum of the two
numbers above.
1
1
2+1= 3
1+3+1= 5
3+4+1= 8
1
8
1
11
© www.teachitmaths.co.uk 2011
1
1
1
7
8
3
4
5
6
21
6
16368
1
4
10
20
35
56
1
3
10
15
28
1
2
15
35
70
1
5
1
6
21
56
1
7
28
1
8
1
12
Make ‘stockings’ by starting
at an edge and kicking down
at the end.
What sequence is created by
summing the numbers in
each row?
Pascal’s triangle
1
1 + 3 + 6 = 10
1 + 5 + 15 + 35 = 56
1 + 6 + 21 = 28
1
1
1
1
4
5
6
7
8
10
15
21
28
© www.teachitmaths.co.uk 2011
6
4
10
20
35
56
1
5
15
35
70
1
21
1st
2nd
3rd
4th
1
7
28
1
8
1
16368
13
1
1
2n - 1
6
56
1
20 = 1
21 = 2
22 = 4
23 = 8
© www.teachitmaths.co.uk 2011
1
1
1
1
6
7
8
1
Can you find other
patterns with different
factors?
1
1
1
21
28
1
1
1
© www.teachitmaths.co.uk 2011
6
7
8
4
21
28
16368
1
3
6
5
15
35
70
6
21
56
16
32
1
1
7
28
1
8
1
16368
14
4
20
1
5
15
35
70
Create a key to translate the
symbols into Arabic numerals (1, 2,
3, 4 ...)
1
10
35
56
20
8
1
1
10
15
10
35
56
1
4
This is the Chinese version of
Pascal's triangle.
2
3
5
3
6
10
4
1
Pascal’s triangle
1
1
3
15
Pascal’s triangle
What do you notice about the
numbers in the red triangle?
They are multiples of 5.
2
4
5
2
1
1
Can you find the nth term?
1
1
Each term is multiplied by 2
to get the next term.
What is the connection
1
1
between the numbers in the
1
2
1
'leg' and the number in the
1
3
3
1
'foot'?
1
Pascal’s triangle
21
56
Explain how the numbers 1-9 are
represented.
1
6
1
7
28
What about higher numbers? How
would you represent 13, 25 or 30?
1
8
1
15
© www.teachitmaths.co.uk 2011
16368
16