For each day in December up to Christmas, can the different tinsel be arranged to join numbers in such a way that they can form equations that equals that day of the month? With twenty five trees to have a go at, this is ideal for working together in groups for a nice pre-Christmas problem-solving challenge.

On the first day of December, nine numbered baubles are placed on a Christmas tree.

To make Christmas mathematically magical, each day, tinsel is placed between the baubles in such a way that each set of numbers can form an equation that equals one – the day of the month. One piece of tinsel cannot intersect another piece. Here is one solution:

1 = 9 – 8

1 = (7 – 6) x (5 – 4)

1 = 3 – 2 x 1

How many solutions are possible? Is there a solution that allows for more colours of tinsel?

On each new day, the tinsel is rearranged to reflect the date. Here is how it could be arranged for the 2nd December:

2 = 9 – 7

2 = (8 – 6 + 4) รท 3

2 = 5 – 2 – 1

Can you arrange the tinsel for each day up to and including Christmas day?

Here is a link to a black and white A3-sized image of all Christmas trees from the 1st December to 25th December and here is a colour version.

There is a lot of things you could vary and focus on.

Which number can have the most pieces of tinsel?

Can any numbers have just one piece of tinsel?

**What if we the order of the numbers in the equation had to be the same order that they appeared on the tinsel?** This likely makes some numbers impossible to complete but that will be part of the exploration in itself.

**Children’s possible use of reasoning skills:**

## Search

The main skill in Searching here is deciding on sets of numbers that will work together. The more numbers they have in a set, the more likely it is they can be used to form an equation of a specific number. That is why it is a nice challenge to try and work out how many pieces of tinsel are possible.

## Organise

There is not an island that emphasises the Organising skill as much. Although you could look at similarities between solutions and the strategies that are used. How can one solution help with another?

## Discover

Whilst not an island that has numerous patterns within, similar strategies can be used between numbers. These can be emphasised using the Discover skills.

They can also conjecture on what they think is the most pieces of tinsel possible for any number and which number it is.

## Investigate

They can continually investigate whether particular numbers of pieces of tinsel are possible as well as the conjectures they come up with.

## Argue

One way to focus on this would be to think about which numbers cannot have a piece of tinsel with just two numbers – prime numbers above 17 and 25 as a square number. As such, these can be trickier to get more pieces of tinsel.

## Explore

In all likelihood, with this one, the exploration of the base set of rules is going to be enough. However, you can shift the focus between different ideas such as how many solutions are possible for particular dates and how many pieces of tinsel are possible.

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