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

What is this problem about?
This is a simple counting problem that can easily be done by being systematic, either by making a list or drawing some pictures.

The fact that the two ice creams are side by side in the cone is important. You get a different result if a blob of ice cream is put on top of another blob. We’ll go into this in more depth in the Solution.

Achievement Objectives
Statistics (Level 2)
- compare familiar or imaginary, but related, events and order them on a scale from least likely to most likely (exploring ways of finding all possible outcomes of simple events by using concrete materials or pictures of objects)

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (systematic list, draw a picture, use equipment).

Resources
Pictures of double headed cones
Blackline master of the problem

Specific learning outcomes
The children will be able to:
- find all the possible outcomes of a simple event using a problem solving strategy (draw, act with objects, list)

1. Interest the children in the problem by discussing favourite ice-cream flavours – you could do a quick tally chart of favourites.
2. Pose the problem to the children – remember to point out that the ice-creams have scoops that are side-by-side (not stacked).
3. Brainstorm for ways to solve the problem.
4. As the children work on the problem (in pairs) ask questions that focus the children 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?
5. Share solutions. Discuss the different ways that have been used to find all the outcomes.

Variations to the problem
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. You might even allow the children to choose the same flavour twice in the same cone.

Extension to the problem
How many different ice creams can you buy if the 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. You might even allow the children to choose the same flavour twice in the same cone.

Solution
Suppose the flavours are vanilla (V), chocolate chip (C), strawberry (B), and passion fruit (P). We’ll now make a list to show all possible cones. Just remember that we have to use different flavours in each cone and that 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. So here they are.

VC     VB     VP     CB      CP     BP

There are 6 possibilities here.

Extension: We’ll do this in the same way. Remember again that we have to use different flavours in each cone but that this time, the flavours sit on top of each other. This means that vanilla on top of chocolate chip is not the same as chocolate chip on top of vanilla. So what do I get this time?

C     B     P     V     B     P
V     V     V     C     C    C

V     C     P    V     C     B
B     B     B     P     P     P

This gives us 12 possibilities.

Now maybe if you have a clever class, they will notice that 12 is twice 6. Is there a reason for this? Certainly. Let’s do the list again in a different order.

C     V     B     V     P     V
V     C     V     B     V     P

   VC          VB           VP

C     B     C      P      B     P
B     C     P      C      P     B

  CB          CP           BP