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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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3388

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

Peaches Today, Peaches Tomorrow...

Age 11 to 14 Challenge Level:

Why do this problem?

This problem is in three parts, with each part becoming more open-ended and requiring more reasoning, giving students a chance to develop their skills at solving problems with fractions and then applying those skills in a more challenging context. By listening to others' approaches, students will be encouraged to persevere and continue to improve on their solution to the final part of the problem.

This task has a playful context which we hope will draw curious learners in, and then offers lots of opportunities to practise routine fraction calculations while making progress towards solving the problem.


Possible approach

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 slides (available as PowerPoint or PDF):  Peaches  Peaches

You may also find this printable worksheet useful: Peaches Worksheet

Once students have had a chance to work on the first challenge, share strategies and discuss any difficulties that arose.

Next, introduce the second task. This is more challenging, as students are given the fractions but not told which order to use them in. While they are working, circulate and look out for good examples of strategic thinking to share with the rest of the class. For example:

The denominator of the fraction must be a factor of the number of peaches
We started with 75 peaches so there are only two options for the first fraction
We could write each fraction on a piece of paper and then move them around to find an order that works!

 
Once each group of students is convinced they have found the correct order, they can make a start on the third challenge. They could start by trying to find a number of peaches less than 100 and a set of fractions so that after a week there is still at least one peach remaining. This 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.

Key questions

How will you record your work efficiently so that you can keep track of what is happening?
 

Possible extension

In part (iii) a solution can be found where the peaches last for more than a fortnight! 
Proving that they have found the longest possible chain requires some detailed mathematical reasoning.
 
Ben's Game and Fair Shares? are both good follow-up fractions problems.
 

Possible support

Students could be presented with a shorter version of part (ii) of the problem:

A little monkey had 44 peaches.

Each day, he kept a fraction of his peaches, gave the rest away, and then ate one.
These are the fractions he decided to keep: $$ \frac{1}{2} \qquad \frac{1}{4} \qquad \frac{3}{4} \qquad \frac{3}{5}$$
In which order did he use the fractions so that he was left with just one peach at the end?

 

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.