# Koala Karts

### Task 228 ... Years 2 - 10

#### Summary

This marvellous task, based on the work of Zoltan Dienes, provides concrete, visual experience with sorting and classifying a set of objects based on their attributes and simple sorting rules. The large cards make roads and the small cards are road signs placed at each forward junction to tell the 'traveller' which way to go. Using the cart, the koalas travel the road system one at a time. The challenge is to be able to describe the attributes of the koalas delivered to the end of the top road.

As the task unfolds the objective is further refined to 'describe using minimum language' which introduces the use of the logical connectives 'and', 'or' and 'not'. As experience grows, students are encouraged to predict then check the attributes of the koalas at the end of the top road in any system.

#### Materials

• 1 cart
• 12 koalas made up of 3 sizes and 4 colours
• 4 road cards and 14 'road sign' cards

#### Content

• reasoning
• recording mathematics
• shapes, properties
• sorting, classifying, ordering

#### Iceberg

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

Mathematical Logic is a particular branch of mathematics and this task introduces students to some of the basic ideas from this study. Consistent with a multiple intelligences approach the ideas of the logical connectives 'and', 'or' and 'not' are introduced in a concrete, visual manner. As students create their own 'logic roads' the emphasis is on discussion and precise use of language.

When a road is labelled with a particular card, all the koalas with that attribute must travel on the labelled fork (at least until the next forward junction). Also, all the ones that do not have that characteristic must travel the other fork. The first question on the card is based on just one road section:

With the blue card in the position shown, only blue koalas can collect on the top road. If the blue card is on the other fork, one could say that red, yellow and green koalas of all sizes collect on the top road; or one could be more precise and say that 'not blue' koalas collect on the top road.

In Question 3, blue koalas travel the full length of the top road, but at the next fork they are joined by any small koalas that have been sent up from the bottom road:

Hence this road network collects koalas which are blue or small on the top road. However, if the road network looked like this:
then the koalas collecting at the top road would be the large and medium blue ones, or more precisely, blue, not small koalas.
• How could road segments and cards be arranged so that the koalas collected at the top road were blue AND small?
• How many such koalas are there in the set?
As the students move on with the task and begin to create their own roads and sorting conditions, encourage them to use only the logical conjunctives AND, OR and NOT to uniquely describe the koalas collected at the end of the top road. Also, encourage them to look for alternative ways to solve each challenge. This can lead to discovering the equivalence of roads which collect:
• large red koalas and small red koalas
• red and large or small koalas
• red and not medium koalas

#### Extensions

• For each road network, can you choose road signs which bring all of the koalas to the end of the top road?
• For each road network, can you choose road signs which bring none of the koalas to the end of the top road? (Mathematically this result is the empty set.)
• If you have other materials with a range of attributes, for example Attribute (or Logic) Blocks, you can make extra road signs and use the same road pieces to extend the complexity of the investigation.
• Consider delivering objects other than koalas and changing the condition cards appropriately. For example the cart might be used to deliver quadrilaterals with the condition cards changed to statements like:
• opposite sides parallel
• opposite sides equal
• opposite sides right angles
• Another example might be to deliver numbers in the cart with conditions like:
• divisible by three
• greater than 100
• between 1 and 2

#### 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.

Many schools have sets of attribute materials. They may not be coloured koalas, but they will have attributes such as shape, colour, size and thickness. If not, such attribute, or logic, sets are easily obtained from education suppliers. Even a collection of buttons could be used - 3 colours and two shapes or sizes is a starting point. (See Task 74, Button Sort.) The roads and road signs used in the task can easily be designed and printed for each group. The rest of the lesson could be guided by a set of images and challenges projected from your computer, interspersed with appropriate discussion and recording.

At this stage, Koala Karts does not have a matching lesson on Maths300, but there are several lessons, such as Lesson 46, Police Line Up, Lesson 47, Farmyard Friends, and Lesson 190, Who Owns The Monkey? which emphasise 'content free' logical reasoning.

#### 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 Koala Karts task is an integral part of:

• MWA Space & Logic Years 5 & 6
• MWA Space & Logic Years 9 & 10