# Interactive Spinners

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

## Problem

*Interactive Spinners printable sheet*

We have two spinners each numbered $1$ to $4$.

We're going to spin them both and add the numbers.

What possible totals might you get?

What is the least likely/most likely total to occur?

Can you summarise your ideas clearly? Can you justify them?

Use the interactivity to test these hypotheses.

Now use the purple cog, in the top right-hand corner of the interactivity, to make a new set of spinners, and write down a hypothesis about the possible outcomes and the shape of the relative frequency bar chart. Justify your ideas to a friend, before testing them with the interactivity.

Once you have experimented with the interactivity you might like to try this problem: Which Spinners?

## Getting Started

What do the experimental outcomes tell you?

Try creating a sample space diagram.

Try creating a sample space diagram.

## Teachers' Resources

### Why do this problem?

This
interactivity offers opportunities for stating and testing
hypotheses, and provides a good chance to introduce sample space
diagrams and to discuss experimental and theoretical
probabilities.

### Possible approach

This lesson would work well in a classroom with a single
computer, which can be displayed for all to see.

Before considering combinations of spinners, introduce the
interactivity with a single spinner, one spin at a time. Clarify
that it is not a frequency table, it's a relative frequency table.
Ask students to describe what will happen for various possible
outcomes of the next spin, then spin to see if they are right.
Repeat until students seem secure with the probability, the
fraction/decimal links and the interactivity. Discuss what they
expect and then spin $100$ (or $50$ $000$) times

Once students are good at predicting what happens when they
have a single spinner, ask them to suggest what will happen if the
interactivity is set to run two identical 'size $6$' spinners. Ask
them to identify possible outcomes and sketch / predict what the
relative frequency bar chart will look like after many spins. Then
run the interactivity for a few $100$ spins.

"Surprised?"

Allow students some thinking and discussion time in pairs
before bringing them together to explain why the bars aren't all of
the same height. Introduce students to sample space diagrams.

Now ask students to apply this technique to analyse the case
of adding two spinners, sizes $3$ and $9$, and draw a good bar
chart to describe the outcomes. If this is done on loose sheets of
paper, the different suggestions can be displayed and compared.
Encourage discussion of the alternatives before seeing the computer
generated bar chart.

The computer also does differences of two spinners. In pairs,
choose two spinners, and work out what the bar chart will look
like. Find a pair who have chosen different spinners to yours, and
convince them that your predicted chart is correct. If they are
convinced, check with the interactivity on the computer, before
choosing a harder example to work on.

### Key questions

Which combinations are most likely? Which are least
likely?

How many times do you think you should run the experiment to
get a reasonable approximation to the real probability?

How can you find an exact value for a probability?

### Possible extension

Encourage students to vary the number of sectors on the
spinners, creating and testing hypotheses of their own based on
these cases.

Once they have experimented with the interactivity, suggest
they take a look at Which
Spinners? where the charts have been produced, and students
must work out which spinners were used.

### Possible support

Having a blank sample space diagram or a blank tree diagram can make this more accessible.What happens if the spinners have just two or three numbers? Can the students work out the theoretical probabilities for these cases?