Make A Snake

Task 5 ... Years 4 - 10

Summary

Students recreate the pattern of coloured rings on the body of Mungo the Maths Snake by using coloured beads or blocks in a defined order. Mungo is born with only one colour ring (Colour A), but each season she sheds her skin and replaces Colour A with an ABA pattern. The visual patterns in the problem can be difficult to 'get hold of' because they stretch out a long way quite quickly, but once key elements are noticed, they can help to identify and explain the significant number patterns in the problem.

Mathematically, the problem is similar to The Mushroom Hunt and Tower of Hanoi.

Materials

Beads in 2 colours and a length of string or small blocks in 2 colours

Content

making and recording patterns
counting and recording numbers
looking for patterns
making predictions and generalisations
powers of 2
symbolic representation
graphical representation of algebraic relationships

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

If using beads and string to make the snake, some students often find that threading on and off is annoying, so they ignore the string and simply line up the colours. This is quite effective and allows for a physical 'substitution' process. Whichever way it is used, the equipment invites students to begin by making the first few seasons. However, the rules of exchange can be difficult to keep under control. One observation which helps is to realise that since:

Mungo starts with Colour A, and
Colour A changes to two of itself with Colour B always in the middle,

then the colour sequence must always alternate and start and finish with Colour A.

As students construct their snakes it seems natural to collect information about the colours and the total. One way to record this is in a table:

Season	Colour A	Colour B	Total
0	1	0	1
1	2	1	3
2	4	3	7
3	8	7	15

Patterns in this table encourage prediction of further growth which can be confirmed, at least for Seasons 4 and 5, by construction. After 10 seasons (Season 9 in the table because we start at zero), the total number of rings will be 1023 and the colours will be made of two consecutive numbers that sum to 1023, that is, 512 for Colour A and 511 for Colour B. Interestingly older students might make such a calculation by application of index laws, but younger children can still access the arithmetic using doubling, perhaps with the help of a calculator.

But a mathematician might ask why these patterns occur in the problem. How do they relate to the rules of the problem? Encouraging students to explain in words (and diagrams) in their journal can lead to recognising that:

The growth rule means that Colour A must double each year.
Colour B counts the total of rings from the previous year because the growth rule means that each each A adds one new B (aBa) and the Bs already present remain.

Students might also reason that since the colours must alternate and start and finish with A, Colour B must always be one less than Colour A.

There is more to discover too, for example that 2⁰ must be equal to 1 if the pattern in the Colour A table is to be consistent. And that, combined with the recognition that B must be one less than A, might lead to a different explanation for the total, such as:

To find the total raise 2 to the power of the year, double this then subtract 1.

Extensions

Graph the pairs (Season, Total) in the table.
Graph the pairs (Season, Colour A) and (Season, Colour B) on the same graph and explore how they sum to make the previous graph.
If I tell you any number of seasons can you tell me the pattern on Mungo's body?
If I tell you the total or the number of As and Bs, can you tell me Mungo's age in seasons?
What happens if we change the rule governing Mungo's colour changes?

Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
Each pair (or group of four) needs two colours of counters, beads or blocks. Since the problem refers to Colours A and B it doesn't matter it is not necessary for different groups to have the same colours. However if you have Poly Plug the yellow/blue plugs are perfect. You might plug into the red board to show the seasons, but you will soon run out of board space.
You might like to tell the story of Mungo's growth using this image of Mungo in the background on your interactive whiteboard. Start the problem by demonstrating the growth rules at a central space, and ask students to go on building 'seasons of Mungo'.

Don't stop just because you reach the end
of the board.
That's better.

Do you notice the other seasons have a yellow
on the end? Why doesn't this one?
We could check it another way using
a drawing.

Encourage recording the number of each colour per season and looking for a way of predicting the number of each colour, and the total, for the next season. Continue the lesson by exploring the ideas above. Be alert for student insights as there are several ways to look at this problem.
At this stage, Make A Snake does not have a matching lesson on Maths300.
Visit Make A Snake in Menu Maths Pack A.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Make A Snake task is an integral part of:

MWA Pattern & Algebra Years 3 & 4

MWA Number & Computation Years 9 & 10

This task is also included in the Task Centre Kit for Aboriginal Students.

Follow this link to Task Centre Home page.