# Crossing The Desert

### Task 94 ... Years 4 - 8

#### Summary

A message has to be taken across a desert and the messenger must return with an answer. However one person can't carry enough food for the full journey and two people can't carry enough for both to do the return journey. The card tells us they can bury food on the way out to be used on the way back, but on which day? It has to be just the right moment so that the messenger will have enough food to go the full journey and so the food carrier has just enough to get back.

#### Materials

• 24 counters & 2 blocks
• Playing board

#### Content

• application of problem solving strategies
• mathematical conversation

#### Iceberg

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

This task is sure to generate discussion.

Students at East Devonport Primary.
Their teacher has organised Mathematician Teams.

There are subtle elements of the problem that must be explored. A mathematician needs to understand, in fact, in many cases clarify and refine, the problem before attempting a solution. Usually this process involves 'talking it through' with colleagues. In this sense the problem encourages justification of answers against agreed criteria, rather than production of the solution, or sequence of solutions.

Discussion will identify some or all of the following aspects of the problem:

• Travelling to the oasis requires 8 days food each way per person and one more food portion for Day 9.
• That is 17 food portions per person, so 34 for two people if they both reach the oasis and return. Therefore the 24 food portions allowed are not enough for both travellers to reach the oasis and return.
• One person will need 17 food portions (the Messenger) so there are 7 others to 'play with'.
• We could assume that each person eats at noon.
• We could assume that the message is delivered 'in seconds' and the Messenger then turns for home. They don't spend a day resting at the oasis and therefore only need one portion of food on Day 9.
• In a similar way, we could also assume that on the day one person (the Companion) turns around, they also only need one portion of food.
Since the card suggests burying food, students usually try that option in conjunction with the trial and improve strategy. However burying alone doesn't reveal a solution; although burying on Day 5 comes close and an interpretation based on this potential solution is made below. The key is to realise that the Food Carrier needs food for their own return journey, could bury some for the Messenger to use on the way back and could give some to the Messenger to make their total back up to the maximum of 12. This understanding leads to the following solutions:

 Day Give Bury Solution 1 3 3 4 Solution 2 4 4 1 Solution 3 4 3 2 Solution 4 4 2 3 Solution 5 4 1 4

Notes

• Solution 1 gets the messenger home with food to spare.
• It could be reasoned that more solutions are possible if you allow progressive burial and/or giving. For example in Solution 5, the four buried portions could be buried 1 on each day, or 2 on Day 2 and 2 on Day 4 and so on. Similarly the giving of 1 could occur on any of the four days. Students can make the decision whether these are seen as merely variations, or as new solutions. It really doesn't matter which. What matters is the reasoned discussion and agreement should the issue arise.
• Equally the students could debate the merits of this potential Bury Only solution. If you accept that in real life the food you eat on Day N provides the energy to travel on Day (N + 1), then they could travel together to Day 5. The Companion would need only 3 pieces of food for the return journey because the third of these would provide the energy to travel home on the last day (Day 1 on the board). The Companion's remaining 4 food portions would be buried. The Messenger would continue the journey, and be able to return to Day 5 on the strength of the food eaten the day before. The four buried portions would be sufficient (consistent with this interpretation) to complete the return journey. Again the real issue is not whether this is a solution, but rather whether it is a solution in the context of this interpretation.
• After many classroom trials, Newcomb Secondary College has suggested that the board provided with the task is better in A4 size. A reproducible version in that size is provided here. It can also serve as a recording sheet.

The problem can be extended further by asking what happens if we change the key numbers? That is, suppose crossing the desert takes a different number of days and the travellers can carry a different amount of food? Can the students find another combination of numbers which also make an interesting problem?

#### 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.

This task works well in groups of three or four. Use the A4 size board provided above and perhaps enlarge it to A3. One large board per group focuses discussion. You will need 24 counters per group but if you have Poly Plug, the red board immediately provides those. The travellers who carry the food can be represented by stacking the 'food' on a small piece of paper that slides across the desert from day to day, or simply by piles of counters manipulated by two students.

At this stage, Crossing The Desert does not have a matching lesson on Maths300. However, for more information about this task being used as a whole class investigation in a Year 1/2 as part of a school's Working Mathematically with Infants kit, visit Shane Hoffman's story in Calculating Changes.

Visit Crossing The Desert in Menu Maths Pack B.

#### 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 Crossing The Desert task is an integral part of:

• MWA Space & Logic Years 5 & 6
• MWA Space & Logic Years 7 & 8

This task is also included in the Primary Library Kit. Solutions for tasks in the latter kit can be found here.