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# Tree Tops

### Why do this problem?

Possible approach

Key questions

### Possible support

### Possible extension

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Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem offers opportunities for students to interpret data in a real life context and process information in order to answer questions. There is quite a lot of freedom in how to approach the task, so students will need to make decisions on how to organise their working.

As part of the "Evaluating Methods, Improving Solutions" feature, this problem encourages students to find a value for the profit after a certain number of years, and then look for ways of improving on it by working within the constraints of the situation. We hope this challenge will engage students' curiosity and encourage them to persevere until they find the optimum
solution.

Possible approach

Set the scene for the problem:

"Imagine you have a large plot of land ideal for planting a forest. First you plant the trees. Then, after 10 years, the forest is thinned to give the remaining trees more room to grow. The forest needs to be thinned again after another 10 years. The wood from the thinning is sold. Finally, the time will come when you cut down all the trees in the forest and sell the timber. You
might decide to do this 30 or 40 years or even longer after first planting your trees! In this lesson, we're going to try to find the best strategy for maximising profits for your forestry company."

Hand out the resource sheet (perhaps cut into three separate sections - Terry's Trees, Theo's Thinning Company, and Lou's Lumber) and invite students to think about the following:

"What sort of questions occur to you that it might be useful to answer in order to find a good strategy?" Give them some time to discuss with their partner, and then bring the class together and write up the questions on the board. Here are some examples of the sort of questions that might emerge:

Which tree is the cheapest to plant?

Which tree minimises the loss after twenty years?

What would the profit be for each type of tree after 30, 40, and 50 years?

What happens to the prices paid for each type of wood over the 100 year period? Why might this be the case?

What's the maximum profit you could make after 100 years?

Then set the class to work on some of their questions. Some students may wish to draw tables by hand and use calculators. Others might prefer to use a computer and create a spreadsheet with the data, and use formulae to calculate the profit for each type of tree and each timescale. As the class are working, circulate to see the different approaches people are using and the different strategies
they come up with for organising the data. It might be appropriate to bring the class together to share some of these strategies, so that students can learn from each other's approaches.

At the end of the lesson, set aside some time for students to explain the strategy they would recommend for planting the forest, together with their reasoning.

Key questions

Does it help to put the information in a table?

Are there any diagrams or graphs that could be used?

Is a spreadsheet useful?

The Getting Started section goes through the calculations for the first forty years for the Sitka Spruce. Students could use this example to repeat the calculations with the other two tree types, and make comparisons.

Students might begin by exploring each type of tree separately. Encourage them to consider the possibility of combining trees - for example, if they try to maximise profits over 100 years, they could plant one type of tree for 60 years and then a different type for the next 40 years. Students could then work towards a strategy for managing a forest with a cycle of planting, thinning and harvesting, that would give a sustainable and regular profit over time.

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.