# 2.3 Changing numbers

If, instead of looking at the shapes, you focused on the number of matches, you probably noticed that the terms went up the same amount each time, in threes. Sequences that have a constant difference have two names (teachers need to know these but learners do not). They are called **linear** or **arithmetic sequences**.

For linear sequences, it is possible to work only with the numbers and ignore the shapes, and some teachers prefer to work like this. They rely on learners remembering a procedure:

- Remember that the 3 times table increases in threes.
- Add a row in the table for the multiples of 3. This row has the rule 3
*n*. - Work out what the change is between 3
*n*and the sequence you want. Adapt the rule to include this change

Let’s demonstrate this procedure.

Step 1 and 2

Position | 1 | 2 | 3 | 4 | 10 | 26 |

Number of matches | 7 | 10 | 13 | 16 | ||

3n | 3 | 6 | 9 | 12 | 30 | 78 |

Step 3

Looking at the third and the second rows (e.g. look at 3 and 7, or 6 and 10), the difference between them is + 4, so we write the rule as

number of matches = 3*n* + 4

This helps complete the table for the 10^{th} and 26^{th} terms.

Should teachers work with the shapes or the numbers? The advantage of working only with number patterns is speed. It provides a relatively quick method for finding a position-to-term rule for a given linear sequence. On the other hand, it is relatively common for learners to misapply this procedure and give the answer 3*n* – 4. Another disadvantage is the reduced opportunity for thinking algebraically since learners tend to focus only on the first term of the sequence (so they are not working with indeterminate quantities). In addition, simply finding a position-to-term rule is a skill that is worth a few marks in examinations, but is not used directly to support other topics. If learners can use the numerical method without writing an extra row in the table, it is a good indication of fluent thinking with symbols. If they cannot, the teacher needs to balance the effects on their learners of spending time memorising and perfecting the procedure, or of trying to make sense of change within a context.