# The Mushroom Hunt

### Task 38 ... Years 4 - 10

#### Summary

Six persons collect mushrooms and on return compare the numbers in their baskets. They notice two special things:
• The total number of mushrooms collected is 63.
• By combining the numbers in their baskets in different ways they can make every number up to 63.
The challenge is to find the numbers that must have been in each basket.

#### Materials

• At least 63 mushroom pegs

#### Content

• doubling
• powers of 2
• binary numbers
• multiplicative (exponential) growth
• non-linear graphs
• problem solving strategies
• reasoning and justification

#### Iceberg

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

This journal page from an unnamed Year 3 student at Crawley Ridge Junior School, Surrey, confirms that even young children enjoy chasing patterns when they 'own' a problem. This student is not far from being interested in converting these answers to binary numbers which record the baskets used to make a solution.

Some find it a little difficult to interpret this problem. You may need to offer additional verbal explanation. The key is that no mushrooms are taken out of the baskets when the persons compare. The combinations are made by counting what's in single baskets, pairs of baskets, triples of baskets and so on up to the total in all six baskets.

• One approach to getting started is to realise that if all the numbers from 1 to 63 are to be made, then one of the baskets must contain just 1 mushroom.
• So, how do you create the number two? It could be that there are two baskets with one mushroom each, or there is one basket with 2 mushrooms.
• The advantage of choosing the option of one basket with 2 mushrooms is that combining it with the basket of 1 mushroom creates a subtotal of 3 mushrooms.
• So, how do you create the number four?
This approach involves breaking the problem into smaller parts. Eventually the students will discover a doubling pattern, ie: 1, 2, 4, 8...
• What happens if seven persons went mushrooming and collect amounts in the same way? Could all totals be made from 1 to ...?
• Suppose all totals could be made from 1 to 32767. How many persons went mushrooming and how many mushrooms were in each basket?
• The baskets and their number of mushrooms can be arranged as ordered pairs, (basket number first, mushrooms in basket second)
(1, 1), (2, 2), (3, 4), (4, 8)...
What happens if those pairs are plotted?
Another way to code the numbers up to 63 is to use a 1 to indicate if a basket is used in creating a number and a 0 to indicate if it is not. For example:
These could be called basket numbers (but mathematicians call them binary numbers).
• Choose your own subtotal numbers and work out their basket equivalent.
• Choose your own basket numbers and work out their equivalent subtotal.

#### 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 convert this task you need heaps of objects to use as mushrooms, and something like paper plates to use as baskets. Poly Plug offers an excellent collection of mushrooms. Probably using the red ones is the simplest approach. A detailed lesson plan for Mushroom Hunt can be found on Maths300.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 130, The Mushroom Hunt.

Visit The Mushroom Hunt on Poly Plug & Tasks.

#### 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 Mushroom Hunt task is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 7 & 8

The Mushroom Hunt lesson is not included in any MWA kit (it was developed after they were published), but it could be used to enhance the Years 9 & 10 Number & Computation kit.

This task is also included in the Task Centre Kit for Aboriginal Students.