# The Farmer's Puzzles

### Task 29 ... Years 2 - 7

#### Summary

A spatial task with a hint of algebra and a reminder of the properties of regular shapes. The farmer puts animals into pens according to certain rules. How does she do it?

#### Materials

• 3 plastic animals of one type and 6 of another
• pop sticks to use as fence pieces

#### Content

• properties of squares, triangles and hexagons
• spatial perception
• spatial and number patterns leading to algebraic generalisation #### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. Not quite... Almost... Aha!!!

Students will soon discover that making square pens won't work. The diagonal of square is longer than its side, so the sticks can't be placed end to end within the square to make triangular pens. However, triangular pens turns out to be the solution when they are arranged like this: The solution suggests a way of making pens that leads to a pattern. Generalising the pattern leads to algebra.

One unfriendly horse: Two unfriendly horses: Three unfriendly horses: Four unfriendly horses: • If I told you any number of unfriendly horses could you tell me how many lengths of fence?
• If I told you how many lengths of fence, could you tell me how many unfriendly horses I could pen?
• The horses and fence pieces are pairs of numbers that belong together because of the structure of the problem, ie: (1, 3), (2, 5), (3, 7), (4, 9)... If these pairs were plotted on a graph, what would you expect to see?
• Would it make sense (in the context of the problem) to join the dots?
If you follow this path there are other questions which could be asked to extend the algebra which develops from this spatial task. For example, continuing the spatial context of the problem, how many ways can the triangular pens be arranged in each 'unfriendly horse' case above. In each case, which spatial arrangement uses the least number of fence pieces?

An answer to Question 2 on the card is suggested by the solution of Question 1. Two of the trapeziums make a hexagon like this: which uses just 12 lengths the six animals can now wallow in equal areas.

Extensions suggested by this solution include:

• Many classrooms have trapezoidal tables. The designers chose these because they can be rearranged in many ways. Encourage students to explore (with actual tables or on triangle paper) all the arrangements that can be made with 2, 3, 4, 5, 6 trapezoidal tables.
• These isosceles trapezia grow themselves. See John Hibbs Notes from an Inspector's Notebook to follow this up.
• The solution is made from six equilateral triangles. This is one example of a hexiamond. Hexiamonds are made from six equilateral triangles arranged so sides match. How many other hexiamonds can the students find?
• How many different ways can the six animals be arranged in the six pens? How do you know when you have found them all?

#### 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 is an easy task to convert to a whole class lesson. All you need is lots of pop sticks or short straws to be fences and counters to be pretend animals.

At this stage, The Farmer's Puzzles does not have a matching lesson on Maths300. However, there is a totally different lesson titled The Farmer's Puzzle, which models a Poster Problem Clinic.

#### 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 Farmer's Puzzles task is an integral part of:

• MWA Space & Logic Years 5 & 6 