Digging Deeper into … Patterns and Sequences
Category : Uncategorized
This can be a difficult strand unit to track through the 1999 Primary Mathematics Curriculum; in junior and senior infants it is titled Extending patterns, in first and second classes it becomes Extending and using patterns, in third and fourth it is called Number patterns and sequences, and in fifth and sixth classes it morphs into Rules and properties. However, it is always from the strand Algebra (check out the maths curriculum glance cards here for more detail) and a summary of the objectives reveals how pattern is at the heart of the strand unit at every class level:
Junior Infants to Second Class > Algebra > Extending patterns >
- identify, copy and extend patterns (colour, shape, size, number)
- recognise patterns (including odd and even numbers; predict subsequent numbers)
- explore and use patterns in addition facts (1st & 2nd)
Third & Fourth Class > Algebra > Number patterns and sequences >
• explore, recognise and record patterns in number
• explore, extend and describe (explain rule for) sequences
• use patterns as an aid in the memorisation of number facts
Fifth & Sixth Class > Algebra > Rules and properties >
• identify relationships and record verbal and simple symbolic rules for number patterns
Different types of patterns
So then, are patterns and sequences the same thing? Actually, there are two main types of patterns:
- Repeating patterns: repetitions of symbols, shapes, numbers etc., that recur in a specific way.
- Increasing (growing) and decreasing (shrinking) patterns: An ordered set of shapes or numbers that are arranged according to a rule. Typically, sequence is also used to describe an increasing or decreasing pattern, particularly if it is a pattern of numbers.
A repeating pattern should have a clearly identifiable core, i.e. the shortest sequence that repeats. It is a good idea to use the terminology of “core” right from the infant classes so that the children understand what is being asked of them.
Children can often copy patterns without even recognising or identifying the core. However, to become competent in accurately extending repeating patterns, it is vital to identify the core. Ways that the children can become more adept at this include verbalising the pattern out loud (“red, blue, yellow, red, blue, yellow, …”) and/or using concrete materials to model the pattern (see below); in this way it is easier to identify the core of the pattern by breaking it apart and laying it alongside the subsequent parts of the pattern to ensure that they match. This strategy of breaking and matching can also be used to help children check have they extended the pattern correctly. It is also for this reason that cubes and links can often be the easiest concrete materials to use for replicating and extending patterns and are preferable initially to threading beads, pegs on a pegboard etc . These can be used instead when the children are ready to progress to more challenging tasks.
Unlike repeating patterns, sequences are more linear; they tend to increase or decrease in specific ways; thus they are also referred to as increasing and decreasing patterns. The way in which the terms (the individual parts of the sequence) are ordered is governed by a rule. Similar to repeating patterns, the children need to be able to identify this rule in order to extend the sequence. To help the children identify the rule in numerical sequences, they should be encouraged to examine each given term and identify what has happened between it and the next term i.e. did the numbers increase, decrease, by how much etc.? They should then record this (e.g. +2, –3) below and between the terms, as modelled in the Operation Maths Pupil books and Discovery books.
Even when extending sequences in their copies, the children should be encouraged to leave an empty line below the sequences, to allow space for them to write in the differences between the terms.
Once the children are comfortable with sequences that increase/decrease by the same amount each term, they can progress to sequences that increase/decrease in varying but repetitive amounts, e.g. +2, +3, +2, …
Odd and even numbers
Odd and even numbers are an example of an increasing pattern/sequence, as the difference between each term is +2. Most children will find it simple enough to recognise odd and even numbers; typically they will tell you that if a number ends in 2, 4, 6, 8 or zero, is it even and, if it does not, it is odd. It is one thing to identify odd and even numbers in this way, but it’s another thing to visualise the numbers and appreciate why they are odd or even. Using concrete materials or pictorial representations is vital for the children to really develop their number sense
and their appreciation of how odd and even numbers interact.
As well as activities like that shown above, pairwise ten frames (ie ten frames placed vertically, as opposed to the more typical horizontal arrangement) can also be useful to model odd and even numbers. Such concrete or pictorial presentations can then be used to show how the total of an odd number and an odd number will always be even as the non-paired cube of both now join to form a pair. These activities also reinforce how only whole numbers can be classified as odd or even, even though a child may incorrectly assume that 1.2 is even, since it ends in 2.
Identifying patterns in addition and multiplication facts
Through concrete activities and activities using the 100 square, it is hoped that the children begin to appreciate the patterns in number facts and that that various groups of multiples are characterised by certain properties eg.:
- When adding/subtracting 10 on the 100 square the answer is always the number directly below/above the starting number.
- When adding/subtracting 9 on the 100 square the answer is always the number diagonally below left /diagonally above right the starting number.
- The multiples of 10 always end in zero
- The digits in the multiples of 9 always total 9 or a multiple of 9 (e.g. 9 × 11 = 99 and 9 + 9 = 18), etc.
Not only will knowledge like this greatly aid their ability to identify and recall the basic number facts, but it will also improve their ability to check and identify errors in their own, and others’ work (e.g. ‘173 × 5 = 858 … hmm, that can’t be correct because multiples of 5 should always end in 5 or 0; I need to do that again’).
Using T-charts to organise information
In the senior classes, when the rules governing the sequences are becoming more complex and less obvious, T-charts can be a very useful way to organise and present the information. They can be particularly useful to help highlight the patterns and how these patterns are developing (see below). They can also provide the children with a clear way to explain how they see the rule. Therefore, the children should be encouraged to use them as a problem-solving strategy as much as is suitable.