I love rich mathematical tasks. Here is one for all levels of schooling. What do you think?

Background to rich tasks

A rich task is an open-ended task that students can engage with at multiple levels. I use the following information from the nrich website when I am talking to teachers about rich tasks.

Background to Dragonistics data cards

In this task we use our Dragonistics data cards, which are shown here. For a less colourful exercise you could use 24 pieces of card with numbers 1 to 8 on them.

Each dragon has a strength rating of between 1 and 8, shown by the coloured dragon scales on the right-hand side of the card. The distribution of dragon strengths is not uniform, but is clustered around the middle, and depends to a certain extent on other aspects of the dragon, such as their species, gender and behaviour.
The students will already be familiar with the dragon cards, and each group of students has a set of about 24 dragonistics data cards. As there are a total of 288 dragons, each group will have a different set of dragons. Some may or may not have dragons of each strength rating.

The task

A dragon team trainer says that teams of two dragons chosen at random nearly always have a combined strength of between 7 and 11.
Is this true?
Provide evidence to support your conclusion.

Try it yourself

If you do not have any dragons of your own, make up about 20 pieces of card, with the numbers 1 to 8 on them, so you can explore the problem. Like Tracy Zager, we emphasise the necessity of exploring the maths ourselves before the children.

Possible approaches

What is great about this exercise is that you can explore it experimentally or theoretically. It has a low entry point, as encouraged on Youcubed. This is sometimes called “low floor, high ceiling”.For younger children, it is a good start to take pairs of dragons, add their strengths, and write down the answer. Then they need to work out a recording method, possibly a tally table.  You can have discussions about what it means for the dragons to be chosen randomly. You can also discuss what “nearly always” means.
Recently I used this task with a group of ten-year-olds. After they had made an attempt at solving it, I asked what they thought would be the most common team strength, and one said 9 or 10 because it is in the middle. I should have explored this idea further. What I did do, was start working out how many different combinations were possible. It is not possible to have a team of strength 1, and there is only one way to get a team of strength 2. How many ways to get strength 3? By the time we got to strength 6, they could see a pattern, that the number of combinations is one less than the total strength. So then I jumped to the other end of the distribution, asking “What is the strongest team we could possibly get?” As it happened, they did have two dragons of strength 8 in their set of dragons, so they correctly estimated the answer to be 16. So then I asked how many different ways they could get 16, and using their previous rule, they suggested 15 ways.  Then when I asked them to tell me what they were, they realised that there was only one way. From there we started working down the numbers. Unfortunately this was during a holiday programme, so I didn’t have time to pursue this further. However we will be using this exercise in our rural rich maths events.

Lessons to bring out at different levels

There are three main ways to approach this problem. The first is to experiment by randomly taking pairs of dragons, and recording their total strengths. A simple theoretical model involves thinking about all the possible outcomes and seeing what proportion of the outcomes lies between the chosen values. Then a more refined model would take into account the distributions of strengths for the given dragons.  The learners may well come up with some interesting other ways to go about this.

Extension questions

A teacher can encourage further thinking with questions such as:
Would this answer be the same for every group of dragons? Is it possible to find a set of dragons so that the only team strengths are between 5 and 11? What would happen if you had teams of three dragons. Does it make a difference if you select one team at a time, and shuffle, or divide into lots of teams and record, before shuffling? How many different team possibilities are possible? What if you only had green dragons – would this make a difference?

Show them the maths

It is important to point out the mathematical skills they are exercising as they tackle rich tasks. This task improves number skills, encourages persistence and risk-taking, develops communication skills as they are required to justify their conclusion. At higher levels it is helping to develop understanding of probability distributions, and you could also introduce or reinforce the idea of a random variable – in this case the team strength.
It would also be interesting to look at the spread for single dragons, two dragon teams and three dragon teams. With enough repetitions (and at this point a spreadsheet could be handy) the central limit theorem will start to be apparent. As you can see, there is great potential to expand this.

Transferring

We need to look at ways this is also applicable in daily life, and not just for dragon trainers. The same sort of problem would occur if you had people buying different numbers of items, or different weights of suitcases. You might like to think of the combined strengths as similar to total scores in sports events. At higher levels you might discuss the concept of independence.
So rich – so many possibilities! Thoughts?