# Interactive Spinners

### 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?