PATTERNING
A pattern is something that repeats itself again and again, in such a
way that you are able to predict what will come next.
Some patterns are increasing, some are decreasing, and others are
simply repeating a core section.
When a pattern is increasing/decreasing, the pattern that repeats
itself again and again, is the amount it is increasing/decreasing by.
When a pattern is a repeating pattern, a core section of
numbers/shapes/letters is repeated in the same order again and
again.
Examples…
Repeating
ABCABCABCABC
Core is ABC
Increasing
2 5 8 11 14 17
Incr by 3
Decreasing
20 18 16 14 12
Decr by 2
Alternating
4 9 7 12 10 15 + 5, -2
Design
@
@@
@@@
Incr by 1 @
each row
New terms…
In an increasing or decreasing pattern, the constant amount each
number is increasing or decreasing by is referred to as the
“common difference”.
For any pattern, the first number is referred to as the “first term” or
“term 1”, the second number is referred to as the “second pattern”
or “term 2”, etc.
The value of the first, second, etc. term is referred to as the “term
value”.
Example:
1, 4, 7, 10, …
In this pattern, the common difference is 3 (because the numbers
are increasing by 3 each time). The first term (or term 1) is 1, the
second term (or term 2) is 4, etc.
This type of pattern can also be represented in a t-chart format…
Term
1
2
3
4
Value
1
4
7
10
For increasing and decreasing patterns, there is a way to determine
the value of a term not listed (i.e., the 97th term) by finding the
relationship between the term numbers and term values in the
pattern.
In earlier years, you have learned that to “describe” a pattern, you
state the rule. For the pattern above, the rule is “start at 1, add 3
each time”. This describes the pattern.
Now, we are going to describe these types of patterns using a
formula. Here is the basic format for the formula…
[common difference] x [term number] +/- [adjustment]
For the pattern above, we can easily determine that the common
difference is 3, so the first part of our formula is determined… 3t
(this means 3 times any term number you choose). Now we need
to determine the adjustment amount. We will use the first three
pairs of numbers in the t-chart we created for this pattern,
substituting numbers for t each time…
3(1) +/- ? = 1
3(2) +/- ? = 4
3(3) +/- ? = 7
In each case, we can see that we
need to subtract 2 to make the
statements true.
So, our formula for this pattern will be 3t – 2. If we substitute any
term number for t, we can then determine the value of that
particular term number, without having to continue the pattern until
we get there.
For example, if we want to find the 97th term for this pattern above,
we substitute 97 for t, and solve the equation.
97th term = 3t - 2
= 3(97) – 2
= 291 – 2
= 289
In other words, if we were to continue the pattern until we reach the
97th number, it would be 289. Finding the formula can save us a lot
of time!! 
Try to determine the formula for each of the patterns below, and
then use this formula to determine what the 97th term would be.
Remember that the “adjustment” value changes for each pattern.
DESCRIBING PATTERNS
1)
Term Number
1
2
3
4
5
6
Term Value
10
16
22
28
34
40
Term Number
1
2
3
4
5
6
Term Value
0
2
4
6
8
10
2)
6)
Term Number
1
2
3
4
5
6
Term Value
2
11
20
29
38
47
Term Number
1
2
3
4
5
6
Term Value
4
8
12
16
20
24
Term Number
1
2
3
4
5
6
Term Value
8
9
10
11
12
13
7)
3)
8)
Term Number
1
2
3
4
5
6
Term Value
8.5
9
9.5
10
10.5
11
Term Number
1
2
3
4
5
6
Term Value
3
7
11
15
19
23
Term Number
1
2
3
4
5
6
Term Value
3
6
11
18
27
38
Term Value
8
11
14
17
20
23
10)
Term Number
1
2
3
4
5
6
Term Value
1
8
27
64
125
216
4)
9)
5)
Term Number
1
2
3
4
5
6