# Flight Departures

### Task 50 ... Years 2 - 7

#### Summary

An observer gives clues about the take off order of four planes at an airport. The challenge is to decide the actual order in which the planes took off.

#### Materials

• 4 numbered toy planes
Note: The type of planes supplied with the task will depend on availabilty.

#### Content

• logical reasoning
• problem solving strategies
• combination theory

#### Iceberg

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

One approach to this problem is the Guess and Check strategy. It combines students' desire to play with the planes with the process of checking against the clues.

Another approach is to look for a clue which gives significant information and use it to eliminate possibilities. For example, the clue:
One of the planes leaves between Plane 3 and Plane 4
means that the order must be 3 __ 4 or 4 __ 3.

Taking this further, these two arrangements must start at the first or second position because there are only 4 positions involved. So, reading from the left, the solution must come from one of:
3 __ 4 __
4 __ 3 __
__ 3 __ 4
__ 4 __ 3
But Plane 3 is not last, so the fourth of these can't be the solution.

Further, Plane 1 does not leave first, so the third of these would be 2 3 1 4 if it were correct. Does it fit with the other clues? Yes, so we have a solution.

But is it the only solution? Continuing this reasoning process with the first and second of the choices above would be one way to decide. Another would be to use the strategy of try every possible case. There are only 24 possibilities and students could be challenged to list them all.

1 2 3 4 ... 1 2 4 3 ... 1 3 2 4 ... 1 3 4 2 ... 1 4 2 3 ... 1 4 3 2
2 1 3 4 ... 2 1 4 3 ... 2 3 1 4 ... 2 3 4 1 ... 2 4 1 3 ... 2 4 3 1
3 1 2 4 ... 3 1 4 2 ... 3 2 1 4 ... 3 2 4 1 ... 3 4 1 2 ... 3 4 2 1
4 1 2 3 ... 4 1 3 2 ... 4 2 1 3 ... 4 2 3 1 ... 4 3 1 2 ... 4 3 2 1
This form of ordered searching is vital to the solution of many mathematical problems. In this case, checking every one of these arrangements against the clues (and deciding whether first is from the left or right), will reveal any and all solutions.

Now the problem gets richer:

• Does the original problem need all the clues?
• Write a different set of clues which gives the same solution as above.
• Choose a take off order and write your own clues for someone else to solve. Make a class set of these student puzzles. Add to the learning by requiring students to use their ICT or creative art skills to produce them. Laminate the results.
• What happens if there are 5 planes?
• Choose a number of planes and write a set of clues which gives exactly 1 ... 2 ... 6 ... 9 solutions?

#### 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.

One way to interest students in this problem as a whole class investigation is to make signs for Planes 1, 2, 3, 4 and ask four students to become the planes. The clues can be printed off for each pair of students (who become air traffic controllers), or displayed on an overhead projector or data projector, or you can divide the class into five groups, give one clue to each group and develop a co-operative exercise to solve the initial problem.

You can also produce printed planes for the students to cut out and use to invesigate the problem in pairs. Once the initial problem is solved it is important to use it to develop the investigation, as in the iceberg suggestions above. In doing so, highlight that this is the way a mathematician works. Refer to the Working Mathematically Process display on the wall, or in the students' journals.

At this stage, Flight Departures does not have a matching lesson on Maths300.

#### 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.

Flight Departures is not in any MWA kit. However it can be used to enrich the Space & Logic kit at Years 3/4 and Years 7/8.