# Farmyard Views

### Task 121 ... Years 2 - 8

#### Summary

A wonderful beginner's task for exploring the same set of objects from looked at from different points of views. This task contributes to the development of spatial perception and includes a challenge to prove the result. The students are encouraged to look at a 2D representation of a farm paddock and imagine themselves in a 3D environment created from it. This skill is similar to that required when interpreting many text book illustrations.

This cameo includes an Investigation Guide.

#### Materials

• 5 cards - large building, small building, tree, person and rooster
• grid matching card sizes

#### Content

• interpreting 2D objects in 3D
• left/right order and in front/behind order
• problem solving skills - including breaking a problem into parts and concept of proof

#### Iceberg

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

A solution to Question 1 is:

 LB P SB T R
Where LB = Large Building, SB = Small Building, P = Person, R = Rooster and T = Tree.

Question 2 asks if there is another solution, but first, the students should be asked to record what would be seen in this case from Sides 3 and 4 (still looking from left to right).

There is only one solution to the given problem, but how can students convince someone of that? Perhaps they will use an argument that breaks the problem into smaller parts such as:

• The first condition could be satisfied by:

 T R SB LB

• But when you move around the corner to Side 2, the only thing you would see is the tree, because the other objects are behind it.
• So T, R & SB would have to move down at least one row to reveal LB.

 LB T R SB

But then in the second row you would see T when you need to see P.

• So T, R & SB have to move to at least the third row and P must be in the second row.

 LB P T R SB

And it must be behind LB because otherwise it would have been seen from Side 1.

• Continuing the reasoning a little further leads to the solution above.
Clearly, although young children will be able to solve the problem and be able to present some form of reasoning that there is no other solution, publishing the reasoning for others to understand places significant demand on communication skills. Publishing in this way is critical in the work of a mathematician. Each new piece of mathematics must be published (a) for colleagues to check and confirm and (b) so it can become a building block in creating further mathematics. For older students, the challenge in this problem is the solution and the presentation of the reasoning that there is no other solution. The presentation does not have to be a written text. This problem may lend itself, for example, to a PowerPoint presentation.

Further, the reasoning above begins with all the objects in the top row, as a solution to the first condition. But there are other ways to arrange the objects to satisfy the first condition, so there is still more to check before it can be concluded with certainty that there is only one solution.

Other iceberg challenges related to this task are:

• Place the five object cards anywhere on the grid and work out a set of clues similar to those in the task. Try them on your partner.
• Repeat with four cards and three cards until you can find a challenge that has at least two solutions.
• Place the tree and the person anywhere you like on the grid. How many ways can you place the other three cards so the rooster cannot be seen from Side 3?
• Or so the rooster cannot be seen and the person can be seen from all sides?

#### 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.

To prepare this task as a whole class investigation you only need to prepare a grid and clip art pictures to match the size of the grid. Or, since the task indicates the value of 'home-made' clip art, use an Investigation Guide that encourages students to create their own drawings. Ownership of materials is one of the features that captivates students as they learn to work like a mathematician.

From here an appropriate start is for each pair to place their pictures anywhere on the grid. Then ask them to imagine a 'walk' around the sides of their paddock from Side 1 and record what can be seen from each side. Teams discuss and check each other's recording. Continue the lesson with the problem on the card and the iceberg aspects appropriate to your year level.

At this stage, Farmyard Views 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.

The Farmyard Views task is an integral part of:

• MWA Space & Logic Years 3 & 4