The games' medals
Problem
Some children from the class are thinking about races because they are interested in the Olympics. Afzal, Bengy and Chrissy like the idea of getting medals. They know that there are Gold, Silver and Bronze medals.
Afzal says that if he won he'd get Gold and if Bengy then came second he'd get Silver and Chrissy would then get Bronze.
Chrissy says that's right but if she'd come first she would get the Gold and then when Bengy came second he'd get Silver again and Afzal would get Bronze.
Bengy has other ideas too.
If the children found all the different ways in which they could be first, second and third, how many would they find?
If we think of Gold first and then Silver and lastly Bronze then so far we have:
Afzal, Bengy, Chrissy
Chrissy, Bengy, Afzal
and whatever Bengy's ideas are.
Which ways can you find?
Getting Started
Who could come first?
How many different people could come first and get the gold medal?
Who could come second?
And third?
So how many different ways of arranging the children will there be?
Student Solutions
We had a good number of ideas sent in for this challenge. New City school London pupils by the names of Patrick, William, Hilin, Jhumana, Anika, Sascha, Rizwan, Joe, Zainab, Aqsa, Bianca, Teddy, Kyra, Shakir, Arif and Hamza all sent in good solutions. Well done all of you! Jiang from the Gardeb International School School in Malaysia sent in the following (their own capital letters)
FIRSTLY,Afzal WOULD LIKE TO GET A GOL MEDAL, SO THE TWO WAYS THAT HE COULD
GET IT IS,
(1.)AFZAL GETS A GOLD,Bengy GETS A SILVER and Chrissy GETS A BRONZE. 2.)AFZAL GETS GOLD, CHRISSY GETS SILVER AND BENGY GETS BRONZE.) SECONDLY, IF BENGY WOULD LIKE TO GET A GOLD MEDAL, THERE ARE TWO WAYS HE COULD GET IT, (1.) BENGY GETS A GOLD, AFZAL GETS A SILVER AND CHRISSY GETS A BRONZE. 2.)BENGY GETS A GOLD, CHRISSY GETS A SILVER AND AFZAL GETS A BRONZE.) THIRDLY, IF CHRISSY WOULD LIKE TO GET A GOLD MEDAL, THERE ARE TWO WAYS HE COULD GET IT,( 1.) CHRISSY GETS A GOLD, AFZAL GETS A SILVER AND BENGY GETS A BRONZE. 3.) CHRISSY GETS A GOLD, BENGY GETS A SILVER AND AFZAL GETS A BRONZE.) OVERALL THERE ARE 6 WAYS THAT EACH OF THEM CAN GET A GOLD MEDAL. I CHOOSE THIS WAY TO SOLVE THIS PROBLEM, BECAUSE BY LISTING EACH WAY ONE BY ONE THE DIFFERENT WAYS THAT A PERSON CAN WIN A GOLD MEDAL, IT IS MUCH EASIER AND CLEARER TO SOLVE.
Another explanation of an idea came from Tanya at Northowram Primary School in which she says;
It says find how many ways of winning is there i found the answer of $6$ ways to win. My aproach: systematically order first all the ways of a winning a,b,c and a,c,b
then all the ways of b winning b,c,a and b,a,c
lastly all the ways of c winning c,b,a and c,a,b
add all the methods together and you get $6$ ways to do it.
We also had good answers from Denika at ACS School in Egham, Mapalo from Sacred Heart School, James and Ben from Milton Rd School, here in Cambridge. Jamal and Holly and Gea sent in solutions but we do not know where from. Well done all of you, you showed such interest.
Teachers' Resources
The Games' Medals
Some children from the class are thinking about races because they are interested in the Olympics. Afzal, Bengy and Chrissy like the idea of getting medals. They know that there are Gold, Silver and Bronze medals.
Afzal says that if he won he'd get Gold and if Bengy then came second he'd get Silver and Chrissy would then get Bronze.
Chrissy says that's right but if she'd come first she would get the Gold and then when Bengy came second he'd get Silver again and Afzal would get Bronze.
Bengy has other ideas too.
If the children found all the different ways in which they could be first, second and third, how many would they find?
If we think of Gold first and then Silver and lastly Bronze then so far we have:
Afzal, Bengy, Chrissy
Chrissy, Bengy, Afzal
and whatever Bengy's ideas are.
Which ways can you find?
Why do this problem?
This activity can be approached practically with young pupils, and it could be used as a starter for class/group discussions. Some children will acquire new mathematical concepts and vocabulary, whilst others will find it helps them to develop skills connected with solving numerical problems.
Possible approach
After telling the story (or your own version appropriate to the children and the classroom context) model the activity with three pupils involved and use the session purely as a discussion in which the children do not necessarily need to do any recording.
Key questions
What other ways of ordering can there be?
Possible extension
Some might like to think about what if there is a "draw" for one of the medals.