How Many Ice Creams?

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?

Teaching sequence

1. Interest the students in the problem by discussing favourite ice-cream flavours – you could do a quick tally chart of favourites.
2. Pose the problem to the students – 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 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?
5. 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 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 students to choose the same flavour twice in the same cone.

Solution

Suppose the flavours are vanilla (V), chocolate (C), strawberry (S), and boysenberry (B). 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     VS     VB     CS      CB     SB

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     S     B     V     S     B
V     V     V     C     C    C

V     C     B    V     C     S
S     S     S     B      B     B

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     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

If you look closely the different cones are occurring in pairs, two for each of the side by side cones.