Here we continue the sharing theme, where the notion of fraction carries two meanings, that of fraction as quotient as well as fraction as parts of a whole. We look at 'equivalent' scenarios, where different numbers of items, shared by different numbers of people, produce the same size share. Though some of the tasks are quite challenging, theses sharing scenarios provide a very accessible model for equivalent fractions.
Task 13A: This can be solved in several ways, including by trial and improvement. For the bottles to be shared out fairly, 6, 9 and 3 bottles should be given to the groups of 10, 15 and 5 hikers respectively. The share that each person gets can be represented by the fractions 6/10, 9/15, 3/5 or, of course, 18/30.
We can think of the 10 people with 6 bottles as being formed of two groups of 5 people with 3 bottles. So the individual shares are the same in the group of 10 and the group of 5 people, and so the fractions 6/10 and 3/5 are equivalent. Similarly, the 15 people with 9 bottles can be thought of as three groups of 5 people with 3 bottles, and so the fractions 9/15 and 3/5 are equivalent. Similarly, the fractions 18/30 and 3/5 are equivalent too.
The task can be solved in a quite grounded way, if pupils take advantage of some of the simpler relations between the numbers of people in the different groups. For example, the group of 15 people consists of half of all the hikers, so they should get half of all the bottles, which is 9. Similarly, the group of 5 people should get one third of this amount (or one 6th of the total amount), namely 3 bottles. And the group of 10 should get twice this, namely 6 bottles.
It might help pupils to spot such relations if the given information is first put in a (ratio) table, as in these examples:
Task 13B: This is more challenging than Task 13A since the two groups will end up with different size shares if they start with full bottles. However, we can make use of the knowledge gained from Task 13A, that ideally everyone should end up with 18/30 or 3/5 of a bottle of water. Can we find shares close to this for each group?
1/9, 17/21; 2/9, 16/21; 3/9. 15/21; 4/9, 14/21; 5/9, 13/21; 6/9, 12/21, 7/9, 11/21; ....
We can reject the first few of these out of hand, and also a pair like 4/9, 14/21 where the first share is less than 1/2.
The most promising pairs seem to be the ones shown in bold. Pupils might be able to spot that 6/9 is slightly more than 3/5=6/10 and that 12/21 is slightly less than 3/5=12/20. It is perhaps less obvious, but for the other pair it is the other way round: 5/9 is slightly less than 3/5=6/10 and 13/21 is slightly more than 3/5=12/20.
But which pair is the fairer of the two? One way to find out is to consider equivalent groups with the same number of people: for example, if we want to compare the fractions (or fair shares) 6/9 and 5/9 with the fraction 3/5, we can consider the fair shares arising from a group with 30 bottles and 45 people or a group with 25 bottles and 45 people, and compare them to a target group with 27 bottles and 45 people; the first group has 3 bottles more, while the second group has only 2 bottles fewer, and so this group's fair shares are closer to the target group's.
A very different approach would be to express the fractions as decimals: Which is closer to 0.6, 0.5 recurring or 0.6 recurring?
Task 13C: Here the fractions work out nicely once more. If the 25 packets are shared fairly between the 35 scouts they get 25/35 of a packet each (or 5/7 of a packet). We can still achieve this by giving 10 packets to the 14 scouts who go outside. Their share is then 10/14 = 5/7 while the share for those left inside is 15/21 which is also equivalent to 5/7.
Task 13D: Again a more challenging variant where we can't quite achieve equal shares across the two groups.
If the 13 packets are shared equally between the 20 guides, they would get 13/20 of a packet each. When they form a groups of 8 and 12 guides, it turns out that the best we can achieve is for the guides who leave the tent to be given 5 packets, which results in equal shares of 5/8 and 8/12 packets in the two groups. The closest alternative is for the group to take 6 packets, leading to shares of 6/8 and 7/12. However, this is not as fair. A nice way to confirm this is to write the fractions 13/20, 5/8 and 6/8 as decimals:
13/20 = 6.5/10 = 0.65; 5/8 = 0.625; 6/8 = 3/4 = 0.75. Pupils who are not familiar with these equivalences could use a calculator, which reinforces the idea of fraction-as-quotient.
Again, we could also compare the fractions 13/20, 5/8 and 6/8 by forming equivalent groups of, say, 40 guides with, respectively, 26, 25 and 30 packets.
Task 13E: Here we broaden things out slightly by moving from 'sharing' to the more general notion of rate. The rate we use is 'packets per day' (but also, just to make life more challenging, days per packet!).
Ella's family consumes 2/7 of a packet per day. John's family takes 3⅓ days per packet.
If we want to compare the families' consumption it is simpler if we use the same rate (packets per day, or days per packet) for the two families, for example 2/7 and 3/10 packets per day for Ella's and John's families respectively. 3/10 is slightly larger than 2/7 (because, for example, 6/20 > 6/21); so John's family eats more cornflakes.
Note that we can make the argument in a more grounded way:
eating 3 packets in 10 days is equivalent to 6 packets in 20 days;
this is a greater rate of consumption than eating 2 packets in 7 days,
which is equivalent to 6 packets in 21 days,
because this means that the same number of packets are eaten over a longer period.
