This problem solving activity has a statistics focus.

The Sloppy Ice Cream Dairy has **four flavours **of ice cream.

How many different cones can you buy that have two different flavours side by side in the cone?

- Find all the possible outcomes of a simple event using a problem solving strategy (draw, act with objects, list).

This is a counting problem that can easily be solved by being systematic. This might involve making a list or drawing some pictures.

The fact that the two ice cream scoops are side by side on the cone is important. You will get a different result if a scoop of ice cream is put on top of another scoop. This becomes evident in the solution.

- Copymaster of the problem (English)
- Copymaster of the problem (Māori)
- Pictures of double headed ice-cream cones

### The Problem

The Sloppy Ice Cream Dairy has four flavours of ice cream. How many different cones can you buy that have two different flavours side by side on a cone?

### Teaching Sequence

- Engage the students in the problem by discussing favourite ice-cream flavours – you could do a quick tally chart of favourites.
- Pose the problem to the students – emphasise that the ice-creams always have two different flavours, and always have scoops that are positioned side-by-side (not stacked).
- Brainstorm ways to solve the problem and discuss how students might keep track of their solutions as they work. Demonstrate these for the class as they are suggested (e.g. drawing pictures, making a list with abbreviations). If students do not suggest making a list, you could suggest and model this.
- As the students work on the problem (in pairs) ask questions that focus the students on ways of counting the outcomes systematically:
*How many different ice-creams have you found?**Have you found them all? How do you know?**How could you convince others that you have found all the ice-creams?* - Share solutions. Discuss the different ways that have been used to find all the outcomes.

#### Extension

How many different ice creams can you buy if the scoops of different flavours are placed one on top of the other in the cone?

It is possible to vary these questions by changing the number of ice cream flavours and by changing the number of different flavours that you can have on each cone. Students may agree they can choose the same flavour twice in the same cone.

### Solution

Suppose the flavours are vanilla (V), chocolate (C), strawberry (S), and boysenberry (B). We must use **different** flavours in each cone and the different types of ice cream are **side by side** on the top of the cone. Remember also that the order of the flavours doesn't matter, for example, VC = CV. These are important pieces of information. Here are the possible solutions.

VC VS VB CS CB SB

There are 6 possibilities here.

#### Solution to the Extension

We’ll do this in the same way. Once again we must use **different** flavours in each cone but this time, the flavours sit **on top** of each other. This means that a vanilla scoop on top of chocolate chip scoop (e.g. VC) is not the same as chocolate chip on top of vanilla (CV). There are 12 possibilities.

C S B V S B

V V V C C C

V C B V C S

S S S B B B

Your class may notice that 12 is twice 6. Is there a reason for this? Look at the list again in a different order.

C V S V B V

V C V S V B

VC VS VB

C S C B S B

S C B C B S

CS CB SB

The different cones occur in pairs, two for each of the side by side cones.