# Farmyard

### Task 72 ... Years 2 - 7

#### Summary

Students are asked to construct a plan view of a farm and divide it into paddocks of identical size and shape. This spatial problem, the second part of which is considerably more difficult than the first, then bridges into perimeter and area questions, which can also be related to pattern and algebra, and value relations challenges.

#### Materials

• Assorted sticks of various colours

#### Content

• 2D spatial perception
• measurement of perimeter and area
• number patterns
• generalisation
• value relations #### 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 Question 1 is straightforward and offers the students early success. It also gives reason to call the shape of the original paddock a Trisquare.

The second question may need a hint. If so, you might suggest dividing each square of the Trisquare into 4 smaller squares and removing sticks until there are only 8 left in the middle. Insufficient sticks are supplied for this approach so the students will have to be resourceful. This 'subtractive' approach is consistent with Task 68, Six Square Puzzle, with which students may be familiar.

Another approach might be to ask if the students could make a smaller Trisquare with yellow sticks. In the end they will discover this solution: Extensions

Value Relations

• Use square dot paper to show other ways the Size 2 Trisquare can be divided into paddocks of identical shape and size.
• Find other ways to divide the Size 2 Trisquare into shapes which are not necessarily identical. You must 'join dots'.
• Consider the two questions on the card. If the original Trisquare field was worth \$1,000 (or any other value) what would be the value of each daughter's paddock in Question 1? Can you check your calculation another way? What would be the value of each son's paddock in Question 2? Can you check your calculation another way?
• What would be the value of each paddock if the original field had been divided like these?    • What happens if the original paddock was shaped like a Sphinx?
Perimeter & Area
• The answer to Question 2 on the card shows that 4 Size 1 Trisquares make a Size 2 Trisquare. Work out the perimeter and area of these two Trisquares.
• Question 2 also shows you how to construct the Size 4 Trisquare. Draw the Size 4 on your dot paper and calculate its perimeter and area.
• Predict the perimeter and area of the Size 8 Trisquare. Find a way to check your answer? Can you check it another way?
• What happens if the original paddock was shaped like a Sphinx?
Generalisation
• If I tell you any size of Trisquare can you tell me its area and perimeter?
• If I tell you any size of Trisquare can you tell me the number of Size 1 Trisquares it can be made from?
• What happens if I tell you Size 3? Size 5? Size 100?
• Use these Trisquares, or make your own, to show how a Size 3 Trisquare can be made from Size 1s.
• What happens if the original paddock was shaped like a Sphinx?

#### 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 use the initial task as a whole class investigation, you only need sticks and dot paper. Matches or pop sticks will work, as will any number of things from the art room. To use the extensions to the full it is probably best to have a set of Trisquares. Of course, part of the investigation could be for the kids to make a class set of cardboard Trisquares.

At this stage, Farmyard does not have a matching lesson on Maths300. However, lessons with related content include Sphinx & What's It Worth?. For more information on either of these investigations, open a new browser tab (or page) and visit Maths300 Lesson 25, Sphinx and Maths300 Lesson 99, What's It Worth?.

#### 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.

Farmyard is not in any MWA kit. However it can be used to enrich the Chance & Measurement kit at Years 5/6 and Pattern & Algebra Years 7/8. 