Charitable Pennies
Mo works for the charity and receives the donations from the schools.
Challenge 1
In 2018 the charity received a total of £1440. Mo noticed that their records show that 15 payments were made (1 from each school), payments were received on 8 different dates, and the same total amount was received on each of those 8 dates.
Find two different ways that this could have happened.
Challenge 2
In 2019 the charity again received exactly £1440, Mo noticed that if the 15 payments were arranged in order from smallest to largest, the amounts went up in equal steps.
Find three different ways that this could have happened.
Challenge 3
Mo wondered if in 2020 the patterns they had noticed in 2018 and 2019 could both occur at the same time. In other words that the total received on each of the 8 dates are equal, the 15 different payments go up in equal steps and the total amount received is £1440 again.
In how many ways can this be done? How do you know?
Here is a printable sheet of the activity.
Why do this problem?
This task was created for a final of the Explore Learning Mathematicians' Award, so it is ideal for groups of four pupils working together. In the competition, pupils are assessed on their team work as well as on their problem-solving skills so this task offers opportunities for learners to apply their mathematical knowledge in a challenging context. Lots of perseverence is needed too!
Possible approach
You could begin by reading out the first part of the problem to the group so that everyone knows the context. Next read out the information for Challenge 1. Some learners may well feel as if they are not sure they have understood so give them time to begin to work on the task and talk to a partner about any questions they have. (It would also be helpful for learners to have a copy of this sheet, perhaps one between two, so that they can refer to the information easily as they work.) After a suitable length of time, you may wish to bring everyone together as an opportunity for pairs to ask questions. Rather than answering yourself, invite others to respond if they can, so that the whole group's understanding of the problem is deepened.
Allow more time for pairs to work on Challenge 1, but encourage them to move on to Challenge 2 when they feel they are ready. You may decide that you would like to bring everyone together specifially to talk about their solutions to Challenge 1 before they have chance to get too far through Challenge 2. Rather than give their answers, you could ask learners to describe how they found their two different ways and talk about whether other pairs went about the task in a different way.
As they continue to work, listen out for any misunderstandings or any insights that you feel are worth sharing with the whole class. You could ask pairs who complete all three challenges to create 'hints' for other pairs who are stuck on a particular aspect of the task. Writing hints without giving a solution or answer is difficult, so encourage pairs to think carefully about how they do this.
Key questions
Tell me about what you have done so far.
Why do you think it's important to... (do that)?
What did you do to find... out?
How did you do that calculation?
Possible support
Having calculators available for those who would like to use them will help to ensure that fluency with mental calculation does not get in the way of reasoning.
Possible extension
If learners have only considered whole pounds introduce the idea that this does not have to be so - can they find other solutions for Challenge 2?
If learners have considered pence as well as pounds for the task, can they find all the solutions for Challenge 2 that only use whole pounds?