Peaches in General
It's like 'Peaches Today, Peaches Tomorrow' but interestingly
generalized.
Age
14 to 16
Challenge level
Problem
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There's a problem which goes like this :
A monkey had some peaches.
He ate half of them plus one
more.
On the second day, he ate half of
the rest plus one more.
On the third day, he ate half of
the rest plus one more again.
On the fourth day, he found there
was only one left.
How many did he have at the
beginning?
By the time you reach Stage 4 you will start to feel that problems like this are examples or instances of something more general.
Try this problem as it is and then generalise your solution as far as you think you can.
Using a spreadsheet may be a help, but you can judge that for yourself.
Getting Started
Obviously start by doing the original problem, that's plenty hard enough as a first challenge if you haven't seen anything like it before, but after that ask yourself the question:
- What things might generalise?
Perhaps the fraction taken each day, or the number of days , or the 'plus one more', or something else, but take those possibilities one at a time.
If you use a spreadsheet (and there are really strong reasons for doing that in this type of problem), what do you need to calculate in each column of the sheet? You can have as many columns as you want, so do simple calculations, then calculations further along the row that use those answers, rather than complicated calculations in single cells.
Will you make your first column a number of peaches before any eating happens? It might be better to have the number of peaches in the final column, making your row a calculation trail that deduces the number at the start?
Student Solutions
Here's a spreadsheet file that helps - Peaches Generally
There are three pages or sheets in this file (look for the coloured tabs at the bottom)
The first called 'Forwards' shows the basic Peaches problem and the solution of 22 is picked out in yellow.
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The second sheet is called 'Backwards' which starts with the
end number of peaches and works back to the start number.
This is more efficient. Can you see why ?
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The general process is to have repetitions of two alternating actions. One action is to multiply by a constant amount. In the basic Peaches problem, that multiplier is 0.5, when half the peaches remain. The second action is to add, or subtract, a constant amount. That value is -1 in the original setup.
Experiment varying these two values. You should quickly find some patterns to think about.
What's going on, and why does it happen ?
You may also notice that there isn't really a 'forwards' or 'backwards', just a line of values and a repeated pair of actions which takes you along the line in the direction you choose. Choosing the opposite direction just uses the inverses of those actions and in the opposite order.
Once you have a really good feel for this you might like to find out about Repayment Mortgages.
There's a multiplier, involving the interest rate, and a constant addition, the amount of the loan paid off each month.
That addition is negative because the loan amount, which is the quantity we really care about, is reduced rather tha increased by each monthly payment.
And if you do get that far you'll have seen something very important about mathematics. A monkey eating peaches might seem a rather childish puzzle, but once we start to generalize the problem we find a valuable abstract form which will appear in the most surprising places.
A mathematician's task is to have a good stock of these abstract forms and to know their key properties.
Mathematicians are always looking for new forms and new properties; posing questions around simple situations is a great way to make these discoveries.
Teachers' Resources
Why do this problem?
This an excellent problem through which to give students an opportunity to experience iterative processes. It also moves, by easy stages, into a multi-variable problem, and the value of using a spreadsheet is readily apparent.Possible approach
Starting with the initial given problem discuss solutions found by the group.Invite ideas about possible directions for generalisation, perhaps starting with the easier results like allowing 'plus one more' to become plus two, plus three, and so on.
Clarify what 'result' has actually been discovered for each generalisation and spend plenty of time letting students sense the 'mathematical need' to account for each 'result'. These are good questions to be 'left in the air', allowing students to turn these over in their minds over time.Key questions
- What generalisations are possible ?
- If we explore those one at a time, which shall we take first ?
- What general statement can we now make ?
- Can we justify that ? Explaining why it should be so.
Possible extension
More able students will produce more extended generalisations and have a motivation to account for what is observed, challenging one another to communicate clear explanations or visualisations of the fundamental processes.
Able students will sense the potential power of a spreadsheet and should be encouraged to work collaboratively to become proficient and confident in its use.