Mr Greenwill looked out on his Otago farm and saw rabbits everywhere.

"I reckon there are about 1280 rabbits in that paddock.

They’ve been doubling in number each year for the last seven years", he said.

How many rabbits were in the paddock seven years ago?

This problem is about recognising that if the rabbit numbers doubled this year, then to find the number of rabbits last year, you have to halve the present number.

This ‘undoing’ is an important theme in mathematics. At all levels we look for operations that will take us back to the original position. An operation that will undo another operation is called its * inverse*. So the inverse of doubling is halving. (And the inverse of halving is doubling.) Similarly the inverse of multiplying by 3 is dividing by 3. Multiplying by anything can be undone by dividing by the same amount, so these are the inverses of each other.

It’s often not so easy to see the inverse of a more complicated operation. Suppose that we double a number and add 1. The inverse of this is subtracting 1 and halving.

You might like to think about other operations and what their inverses might be. What is the inverse of adding 5? What is the inverse of multiplying by a half? What is the inverse of rotating about 90 degrees?

Because this is an inverse or undoing problem, a good strategy is to work backwards.

Estimating (half), then using a calculator to check is one approach. Some calculators will repeat an operation by just touching one button.

### The Problem

Mr Greenwill looked out on his Otago farm and saw rabbits every where. "I reckon there are about 1280 rabbits in that paddock. They’ve been doubling in number for the last seven years", he said.

How many rabbits were in the paddock seven years ago?

### Teaching Sequence

- Ask the students to explain in their own words what doubling means.
- Ask the students to double given numbers mentally, for example, 5, 12, 16, then to restate the original number as half of the double.
- Share solutions to the doubling. Ask the students to explain how they doubled. For example some students when doubling 12 will use 12 + 10 + 2, others may use: 10 + 10 + 2 + 2. Ask how they could therefore halve 24.
- Ask student to explain in their own words the connection between doubling and halving. ('They are opposites.' 'One undoes the other.')
- Pose a larger double, for example, 143. Ask the students for possible approaches and results. (If a calculator is suggested, have students estimate first.) Then 'undo' 286 to confirm 143 is half.
- Read the rabbit problem with the class.
- As the students work on the problem ask questions that focus their thinking on the reasonableness of their answers and the appropriate use of calculators.
*How are you solving the problem? Why did you decide to do that?**Is your answer reasonable? Why?**How do you decide when it’s a good idea to use a calculator?**Do you think it’s a good idea to use a calculator for all your problems? Why? Why not?**How do you keep track when using a calculator?* - Share solutions.

#### Other Contexts

This could apply to money or apples on a tree or all sorts of pets, especially mice.

#### Extension to the problem

Suppose the rabbits numbers doubled and had 6 added to them each year. How many rabbits would there be after 5 years if there were 8 rabbits at the start?

Now go back from that. Suppose that the rabbit numbers doubled and had 6 added to them each year. If there were 170 after 4 years, how many had there been at the start?

### Solution

If now there are 1280 rabbits, then a year ago there were 640, two years ago there were 320, three years ago there were 160, four years ago there were 80, five years ago there were 40, six years ago there were 20 and seven years ago there were 10.

So seven years ago there were only 10 rabbits in Mr Greenwill’s paddock.

#### Solution to the extensions

The numbers increase like this: 8; 22; 50; 106; 218; 442.

What do you have to do to ‘undo’ doubling and adding 6? The answer is subtracting six and dividing by two. So going backwards from 170 we get: 82; 38; 16; 5. Four years ago there were only 5 rabbits.