Rabbits in the pen
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
Age
7 to 11
Challenge level
Problem
Image
Let's imagine that rabbits can only be one of these $6$ different kinds and there are no other kinds of rabbits. Then:
How many, and of what kind, will there be in a pen if all these statements are true?
- There are more brown than any other colour
- There are equal numbers of floppy browns and ordinary browns
- There are three which have floppy ears
- There are twice as many floppy browns as floppy whites
- There are three times as many blacks as whites
How many, and of what kind, will there be in the second shop if all these statements are true?
- There are the same number of blacks as browns
- There are twice as many floppy browns as ordinary browns
- There are the same number of floppy browns as floppy greys
- All but one are floppy
Getting Started
How about trying with one floppy brown first. How many of the other types of rabbits would this mean there are?
Does this work? If not, you could try with two floppy browns.
How will you record what you are trying?
You could use counters or blocks for the different kinds of rabbits. Or you could use symbols or simple pictures on paper.
Student Solutions
We had a very large number of solutions sent - many of them were particularly pleasing as they included really good explanations as to how you were thinking. This is something we do wish to encourage.
As well as individual solutions coming in there were quite a number from groups of children at Oxgangs, Hotwells, St. Joseph's and Gateway schools. Thank you to the staff who gave the pupils opportunities to send in solutions. Here is a solution that came from Amanda and Paula.
Solution to the first pen-
First we started with trial and error to find out how many brown floppies
there would be. Then we found out that one wouldn't work so we tried two.
So if there were two brown floppies then there has to be two brown
ordinaries too. Then we wrote that and charted down the rest of the
categories like this-
Brown flop: 2
Brown ordinary: 2
Black flop : ?
Black ordinary: ?
White flop: ?
White ordinary: ?
After that we tried to find out what the white and black were.. So we knew
that there are three times as many blacks as whites. So if we put one next
to the white flop then three next to the black ordinary like this-
Brown flop: 2
Brown ordinary: 2
Black flop: ?
Black ordinary: 3
White flop: 1
White ordinary: ?
And we knew that it couldn't be black flop because if it was then there
are three with floppy ears that wouldn't work. And then we put in zeros
next to the empty ones like this-
Brown flop: 2
Brown ordinary: 2
Black flop: 0
Black ordinary: 3
White flop: 1
White ordinary: 0
And then we checked it and it worked.
Solution to the second pen-
First we did trial and error to try and find out how many floppy browns
there are. Then we started with one and we knew that couldn't be right
because brown ordinary has to be half of brown flop. There can only be one
ordinary so brown flop definitely had to be two. We put zeros next to all
the ordinaries because there can only be one ordinary and that is brown. We
wrote it out like this-
Brown flop: 2
Brown ordinary: 1
Black flop; ?
Black ordinary: 0
Grey flop: ?
Grey ordinary: 0
Then to find out what black flop was we added up both the browns and it
equaled three. We did that because there is the same number as blacks as
browns. We added it in like this-
Brown flop: 2
Brown ordinary: 1
Black flop: 3
Black ordinary: 0
Grey flop: ?
Grey ordinary: 0
So now all we need to work out is grey flop. And we knew that there are
the same number as floppy browns as floppy greys. Floppy brown is two so
grey flop is also two.So we put it like this-
Brown flop: 2
Brown ordinary: 1
Black flop: 3
Black ordinary: 0
Grey flop: 2
Grey ordinary: 0
Then we checked it and we found out it all worked out.
Teachers' Resources
Rabbits in the Pen
Image
Let's imagine that rabbits can only be one of these $6$ different kinds and there are no other kinds of rabbits. Then:
How many, and of what kind, will there be in a pen if all these statements are true?
- There are more brown than any other colour
- There are equal numbers of floppy browns and ordinary browns
- There are three which have floppy ears
- There are twice as many floppy browns as floppy whites
- There are three times as many blacks as whites
How many, and of what kind, will there be in the second shop if all these statements are true?
- There are the same number of blacks as browns
- There are twice as many floppy browns as ordinary browns
- There are the same number of floppy browns as floppy greys
- All but one are floppy
Why do this problem?
This problem is a good context for taking pupils beyond just straightforward arithmetic, by also giving them the opportunity to extend logical thinking skills.
Possible approach
A nice way to lead into the problem would be to show the class the picture of the rabbits in the pen and simply ask them to describe what they see, wihtout saying anything more yourself. Pupils will mention the different colours and different kinds of ears, and many will notice that each rabbit is different from all the others. You can use this to introduce the fact that we only have these
six different kinds of rabbit.
At this point, it would be good to use objects or symbols on the board to represent rabbits. You could print off pictures of each type of rabbit and stick them on the board, or alternatively, you could simply draw circles in the three different colours with lines to represent ears, pointing up or down. Whichever way you choose, start off by drawing just, for example, 1 brown floppy eared, 2
brown ordinary eared and 1 grey floppy eared rabbit. Ask the children to describe what they see now and begin to bring out some numerical comparisons. Some might say "there are the same number of floppy eared rabbits as ordinary eared rabbits" or "there are three times as many brown rabbits as grey rabbits". In this way, the children will become familiar with the language and will be more
confident when tackling the problems.
As they are working on the two different pens in the problem (perhaps in pairs), you might want to have counters, coloured pens etc available for the children to use. You could stop the whole group at a convenient point to discuss how they are tackling the problem. Encourage trial and improvement strategies which involve trying out a small number of one type of rabbit and building up the
other kinds of rabbits from the information given. Then, if that doesn't fit the information, trying one more of the first type of rabbit and so on. You could also invite pupils to share their methods of recording with each other - some may have devised effective short-hand symbols or letters for the rabbit types.
Key questions
How could we start on this challenge? What could we try first?
Does this fit with the other statements? . . . . if not, why not?
What could we try next?
Possible extension
Pupils could set up another situation of their own and make statements that will lead to the unique situation. They could then challenge a friend to use the statements to find out the numbers of each type of rabbit.
Possible support
You may prefer to create some slightly simpler challenges, for example:
- There are twice as many floppy ears as ordinary ears
- There are the same number of grey floppy ears as brown floppy ears and as black floppy ears
- There are the same number of grey ordinary ears as brown ordinary ears and as black ordinary ears
- There are 9 rabbits altogether