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'Peaches Today, Peaches Tomorrow....' printed from http://nrich.maths.org/
This problem could replace repetitive textbook work on calculating fractions of integers. It offers plenty of practice of these calculations while requiring students to come up with problem-solving strategies. It offers a good context for thinking about factors of numbers.
Introduce the first part of the problem to the class, and give them a little bit of time to work in pairs to solve it. All three parts of this problem can be displayed on these PowerPoint slides
Once students have had a chance to work on the first challenge, share strategies and discuss any difficulties that arose.
The second task is much more demanding and interesting, requiring students to work quite systematically and record the steps they take clearly. Challenge students to find the solution and to convince themselves that there is no other way of ordering the fractions.
Finally, the last challenge would work very well as a 'simmering activity' set for students to think about beyond a single lesson - perhaps this could be set as a homework. The best solution could be displayed on the classroom wall, and students could be challenged to improve on it.
How will you record your work efficiently so that you can keep track of what is happening?
In part (iii) a solution can be found where the peaches last for more than a fortnight! Finding a solution of this magnitude should provide a suitable extension challenge.
For part (ii), suggest to students who are struggling to record their thinking effectively that they could use a tree diagram: at each stage, branch off the fractions it would be possible to try next so that all possibilities are checked.