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?

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

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

Pictures of double headed cones

### 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 in the cone?

### Teaching sequence

- Interest 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 – remember to point out that the ice-creams have scoops that are side-by-side (not stacked).
- Brainstorm ways to solve the problem and discuss how students might keep track of their solutions as they work.
- 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 to the problem

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** in 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.

#### 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 is not the same as chocolate chip on top of vanilla. 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 are occurring in pairs, two for each of the side by side cones.