# Cross-bear

The 'Something in Common' collection from John Burke contains problems that enable students to hone their skills of both problem solving and proof. Each investigation comes with several different versions of a problem that have 'something in common'. Having discovered that the problems are linked by 'something in common', an extension task is to prove the links will always work.

The 'Cross-Bear' resource is an interesting variation on 'Goldilocks and the Three Bears'. The story is that Daddy Bear, Mummy Bear and Baby Bear sit down to eat three identical bowls of porridge. They begin eating together, but their table manners are shocking and they will start to eat each other's porridge once they have finished their own!

One example of the problem has the following information:

- Daddy Bear, Mummy Bear and Baby Bear all start eating their porridge at the same time. Daddy Bear finishes his porridge in 50 seconds and then starts eating from Mummy Bear's bowl.
- After a further 10 seconds Mummy Bear's bowl is empty and so Daddy Bear and Mummy Bear start eating from Baby Bear's bowl.
- In a further 15 seconds, Baby Bear's bowl is empty.

The question is, how long would it take each bear to eat one full bowl of porridge?

The image below summarises events.

One method of solving the problem is to let the number of seconds to finish a full bowl of porridge on their own by Daddy Bear, Mummy Bear and Baby Bear be D, M and B respectively.

Daddy Bear finishes his porridge in 50 seconds, hence D = 50

Daddy Bear spends the next 10 seconds helping Mummy Bear to eat her porridge. This means that Mummy Bear spends 50 + 10 = 60 seconds eating most of her porridge, but Daddy Bear also spent 10 seconds eating some of mummy's porridge. We know that Daddy Bear eats at a rate of 1 bowl in 50 seconds.

It is important that students can interpret this result and are clear that this means that mum would take 75 seconds to eat a bowl of porridge on her own without any help from Daddy Bear.

For the final 15 seconds, all three bears are eating from Baby Bear's bowl. Baby Bear eats from its bowl for a total of 50 + 10 + 15 = 75 seconds.

Working through the problem is the first stage. The resource contains sixteen different versions of the problem, so a class working in pairs can each work on what appears to be a different problem. However, collecting together all of the different examples gives the following results:

- Baby Bear takes 10 times the last time interval to eat a bowl of porridge.
- Baby Bear eats half a bowl of porridge.
- Baby Bear is three times slower than Daddy Bear at eating porridge.
- Mummy Bear eats 1.5 times slower than Daddy Bear.

A demanding extension task is to look at how to choose the starting conditions so that the final relationships hold. As an example, how do we determine the time Daddy Bear and Mummy Bear are eating from Mummy's bowl in order for Mummy Bear to eat 1.5 times slower than Daddy for each of the problems?

Let D be the time for Daddy Bear to eat a bowl of porridge.

Let M be the time for Mummy Bear to eat a bowl of porridge.

Let (1/a)D be the time that Daddy Bear and Mummy Bear eat from Mummy Bear's bowl.

Using a = 5 gives the correct relationship that Mummy Bear eats 1.5 times slower than Daddy, and that in order to achieve this the time they both eat from Mummy Bear's bowl is 1/5 of the time it takes Daddy Bear to finish his bowl.

We have recently added the complete collection of resources from John Burke's 'Something in Common' materials, which comprise; investigations, puzzles and other enrichment activities. The activity described here is 'Cross-bear'.

You can also join us at the STEM Centre to experience ideas for teaching through understanding on ENTHUSE funded courses, details of which are here.