From the changing weather to the shape of a cauliflower, we are surrounded by patterns, making their study crucial to maths and children’s understanding of algebra…
Some say that mathematics is all about the study of patterns and if patterns are not the whole of mathematics, then they are a most important part. Mathematically, patterns are a big idea because they are at the heart of algebra, which is just a way of expressing a pattern succinctly. The triangular numbers, for example, are based on the pattern of 1, 1 + 2, 1 + 2 + 3, and so on, a pattern that can be seen by laying out counters:
The tenth triangular number is arrived at by adding together 1, 2, 3 and so on up to 10. While the pattern is easy to see, it is less easy to articulate. How do you describe any triangular number? Well it is the sum of all the whole numbers up to and including that number. A bit of a mouthful. Mathematicians think it is more elegant to describe the nth triangular number in this way.
3rd triangular number: 1 + 2 + 3
7th: 1 + 2 + 3 + 4 + 5 + 6 + 7
nth: 1 + 2 + 3 + 4 ….. + (n-2) +
(n-1) + n.
Conceptually, therefore, working with patterns is important because it provides a sound foundation for thinking about algebra. But it is also important because looking for and finding patterns is at the heart of reasoning mathematically. Take, for example, the patterns in the multiplication tables. Children soon spot there is a pattern to the multiples of 10, in that they always end in zero.
Other patterns often explored include noticing that the digits of any multiple of nine will add to nine (or a multiple of nine) providing a handy check: 468 is a multiple of nine because 4 + 6 + 8 = 18 (and 1 + 8 = 9). Similar patterns hold for multiples of three or six.
A less often explored pattern involves whether a number is divisible by a power of two (2, 4, 8, 16, etc). The test for a leap year is whether the year is a multiple of four, and the quick test for that is whether the last two digits of the year are divisible by four. So 2014 is not a leap year, nor is 2015, but 2016 will be.
Why does that test work? If we think of 2016 as 2000 + 16, then we know four divides exactly into 16. We also know that four divides exactly into 2000 because this is a multiple of 100 (2000 = 20 x 100) and s ince four divides into 100 it must divide into any multiple of 100.
We can think of the test for even numbers in a similar way: 534 is 530 + 4 and since 530 is a multiple of 10, this must be divisible by two – so it only matters if the ones column is divisible by two. So, if the last digit of a number divides by two, the whole number is divisible by two and if the last two digits are divisible by four, so is the whole number – the beginning of a pattern.
The question then is, is there a test for divisibility by eight? Yes, multiples of 1000 are divisible by eight (since 1000 = 10 x 10 x 10 = 2 x 5 x 2 x 5 x 2 x 5 = 2 x 2 x 2 x 5 x 5 x 5 = 8 x 125). It follows that if the last three digits of a number are divisible by eight, the whole number must be. 234640? 640 is a multiple of eight, so 234640 must also be. While you might rarely use this test, the drive to look for patterns is an important disposition to encourage in learners.
Repeating and growing patterns
The patterns most often met in primary maths change in predictable ways, usually as repeating patterns or growing patterns.
Even with simple repeating patterns, children can be challenged to make predictions about what will be, say, the 10th, 100th or 39th element of the pattern. For example, with a simple repeating pattern like:
X X O X X O X X O X X
Children should be able to identify that every third element is O, so the 10th element will be an X (as 10 is not a multiple of three, or, as children are likely to express it, 10 is not in the three times table). Similarly, they will be able to figure out that 100 will also be an X, as 99 is going to be O. And the 39th element will be an O, reasoning along similar lines.
In a growing pattern there is a constant relationship between successive terms. Most of the growing patterns that learners meet in primary school are linear growing patterns, which means terms in the pattern increase or decrease by a constant difference (Yes, in the pattern 10, 7, 4, 1, −2, - 5, … the terms get smaller, but such a pattern comes under the blanket heading of a growing pattern).
While learners can explore growing patterns in purely numerical terms, it helps to provide a context and image for these, so the reasoning that leads to figuring out the value of any term is grounded in these contexts and images. Take, for example, the context of putting out tables in long rows for a banquet. As the number of tables increases, so the number of people who can be seated can be recorded: (Fig.2)
Recording this in a table:
Table: 1 2 3 4 5
People: 4 6 8 10 12
Children have little difficulty noticing that the pattern is ‘going up in twos’ and, asked how many people can sit around a row of 10 tables, may count up in twos from four to get to the answer of 22. But this becomes laborious for, say, 100 tables (the banquet is taking place in Wembley stadium).
Encourage children to articulate the general rule by referring to the context and images. Each table has two people sitting opposite each other so the total number of people is double the number of tables, plus two for the two people at each end. Some learners will be able to express this using algebra: if n stands for the number of tables, then there are 2 x n + 2 people sitting around them.
The other sort of growing pattern is one in which there is a constant ratio between terms – successive terms are related multiplicatively rather than additively. In these sorts of patterns, terms rapidly increase (or decrease) relative to each other, in ways described as exponential. The mathematics here is less often met in primary school, but learners can be introduced to the idea through classic problems, such as the chessboard and rice problem (see The emperor’s rice).
Repeating patterns and geometry
Everyday examples of repeating patterns are all around us: wallpaper friezes, repeating prints on fabrics, the brickwork on buildings. Young children these patterns through activities such as making potato prints. Even something as seemingly simple as this can form the basis of an interesting line of inquiry for older learners. Imagine the repeating pattern is created by a simple footprint. There are essentially three ways that a ‘frieze’ pattern can be created.
Learners can explore the different friezes that can be made using these basic transformations. You may expect there are many, if not an infinite number of such patterns, but there are essentially only seven different underlying patterns (although obviously an infinite number of different motifs).
Learners can search on the internet for the variety of patterns and check if they have all seven (There are some lovely images of cast iron patterns at http://nrich.maths.org/1341). This seemingly simple activity is at the heart of an entire branch of university mathematics.
Early years: clap and tap
Create a simple repeating rhythm using claps and taps (striking both hands on your knees). For example:
Clap, clap, clap, tap, tap, clap, clap, clap, tap, tap, clap, clap, clap, tap, tap,…
Invite children to join in.
After everyone has got the rhythm, set up counting the sounds and actions:
Clap, clap, clap, tap, tap, clap, clap, clap, tap, tap, clap, clap, clap, tap, tap, …
1 2 3 4 5 6 7…
Who can say what sound will be made on the tenth count? The twentieth? The hundredth?
Children go off in pairs and make up their own pattern. Can they find some way to record it on paper so that someone else could carry out the same pattern?
What about adding another sound, say, a stamp?
Provide learners with a copy of the calendar for the month.
Learners put a square box around four dates: two next to each other and the two immediately below.
They add the pairs of numbers diagonally opposite each other. (11 + 19 and 12 + 18 in the example above). What do they notice about the answers?
What happens with another set of four numbers? With a bigger square? With a rectangle?
Set up this linear growing pattern
Can children say how many squares will be in the 10th T, the 20th, the hundredth?
Outstanding schools: RJ Mitchell Primary
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