### Task 122 ... Years 2 - 8

#### Summary

The Australian Football League (AFL) provides the setting for this language and logic puzzle. It doesn't matter if this is not your local code because the logic is still accessible. Later your local sporting code - football or other - can be used to challenge students to make their own puzzles.

The major aspect is the provision of concrete materials (cards) which make the task accessible to, and popular with, almost all students. Another feature is a kinaesthetic option where the class members can 'act out' the ladder positions. Include the way points are assigned for a win, loss or draw and you have another challenge.

An extension of the task offers an excellent way to introduce selections and arrangements (permutations and combinations). The teacher (or the students) can choose the conditions for the problem to manage the difficulty of the challenges.

#### Materials

• 16 cards, each with a team name

#### Content

• problem solving strategies
• mathematical language related to space and order
• number facts and relationships
• 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.

The solution to the puzzle (from the top of the ladder) is:

Hawthorn, Fremantle, Essendon, Carlton, Brisbane, Sydney, Adelaide, Melbourne, Collingwood, Geelong, Richmond, Kangaroos, West Coast, St. Kilda, Port Adelaide, Bulldogs
To support students to discover this, refer to the Working Mathematically process and ask what strategies a mathematician might use for this problem. They have the cards, which enable them to make a model, but what to do now. Guess & check, break the problem into smaller parts and draw a diagram all offer possibilities. One of the smaller parts, for example, is the clue:
West Coast won the same number of games as St. Kilda.
The consequence is that they must be near each other on the ladder.

Extensions

1. Ask the students how they think the person who designed this challenge began to create it. Most likely they began by arranging the list of teams (probably so their team was on the top!). Offer students the opportunity to develop their own Ladder Challenge to add to the developing class set. They will soon discover this is not an easy task because there have to be just enough clues to solve the puzzle, the language needs to be clear and the draft clue set and solution has to be checked by others before it can be considered for the class set.
Note
It is at this point that the logic of the puzzle can be made relevant to the particular class. For example, in 2012 the AFL expanded to 18 teams by including the Gold Coast Suns and the Greater Western Sydney Giants. This will be very relevant to a class supporting the AFL code.
Equally, if the class supports Rugby, or American Grid Iron, or Soccer, making a football ladder puzzle for 'their' code, or for a netball competition, or any other sport with a competitive ladder, will be one of the features that encourages students to be fascinated by this challenge.
2. The order on the AFL ladder is decided by both the number of games won and the points assigned for winning, losing or drawing each game. Challenge students to develop a set of clues that is either totally based on the points each team earns, or at least includes some clues about points.
3. If there were no clues, how many ways could the ladder be arranged?
Answering this requires using logic such as:
There are 16 choices for the first place on the ladder. And for each of these there are 15 choices for the next. And for each of these pairs, there are 14 places for the next ... and so on.
This leads to the calculation 16x15x14x ...3x2x1 = 16!

Once this type of reasoning has been encouraged, suggest students to make a ladder with, say, five teams and calculate the number of possible ladders with no clues. Then they write their first clue and calculate the number of possible solutions, then the next, and so on until the answer to the calculation is 1, which will be the unique solution to the puzzle.

Perhaps this extension is more suitable for older students.

#### 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.

Use a set of cards about 20cm square and write the team names, one per card. This allows the problem to be acted out with 16 class members who each hold a card. Also, the acting out can be confirmed by students folding and tearing scrap paper (two pieces per pair and eight sections in each) to make team cards. Transcribe the clues to the whiteboard or a duplicated sheet (copyright prevents you photocopying the card) and you are away.

Strategies teachers have used with the whole class and these tools include:

• Asking each pair to find the solution and expecting that the class will be responsible for justifying their answer to other teams until there is general agreement about the solution.
• Allowing the students who don't have cards to instruct those who do while the teacher stands back and watches chaos gradually develop into order.
• Handing out clues so that each person not holding a card is responsible for the correct inclusion of one clue, but can't see all the other clues at the same time. They have to rely on others being equally responsible for their clue.
• Handing a set of clues to everyone, perhaps including card holders, and challenging the class to solve the problem in complete silence.
The thrust of the lesson is to highlight the importance of the mathematician's strategy toolbox.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 69, Football Ladder, which includes several clue sets.

#### 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.