2. Magic Loops You will need: A long length of string
A tree, large fence post, chair or other object
Some friends – groups of 4 work best, but 3 would do.

3. Magic Loops Tie a small loop in the end of your string.
Wrap your string around a convenient object, such as a tree trunk, counting off 5 turns.
Tie another small loop in your string.
Wrap your string a further 4 times round your tree trunk and again tie a small loop.
Wrap your string a further 3 times and tie a final small loop. Cut off the remaining string.

4. Magic Loops
Unravel your string from the tree trunk and get your 3 willing helpers to each hold a loop. (Put the two end loops together.)
Each person needs to pull gently against the other 2 people and as they do so, the string forms a triangle.
What do you notice about it?
Why has this particular shape been created?
How accurate is it?

5. Magic Loops Who uses this?
Builders have long known the properties of a 3:4:5 triangle, and some still use a string tied with 12 equally spaced knots to check that walls are perpendicular.

7. Place your book! You will need:
2 markers – which could be rounders posts or 2 students who have promised to stand still…
Everyone else will need themselves and their exercise book (ideally), or a sheet of A4 paper.

8. Place your book!
Place the two rounders posts (students standing still) about metres apart.
All other students do the following:
Find a position where you can place your book on the ground so that extending a line along one edge of it would touch one post and extending a line along the other side would touch the other post.
Stand next to your book.
See diagram on next slide!

9. Place your book! (Aerial view)
Posts
Look along the edges of your book to line it up
Book

10. Place your book! There should be lots of places to stand.
Once everyone is in place, what do you notice?
Why has this happened?

12. How tall is it?
There are several different ways to estimate the heights of trees or buildings. Either each group use one of the methods to estimate the height of several items or each group try 2 or 3 different methods for measuring one item.

13. How tall is it? 1
Using shadows. You need: A metre rule and a long tape measure If you know the length of the shadow of a metre rule, how can you use this to estimate the height of an object based on the length of its shadow?

14. How tall is it? 2
Simple rules You need: A 30cm ruler and a long tape measure Hold the ruler vertically and find the place where it visually touches the top and bottom of the tree. Take the measurements shown on the next slide and use them to find the height of the tree.

15. How tall is it? 2
Measure these 3

16. How tall is it? 3
Get an angle You need: A clinometer and a long tape measure Take the measurements shown on the next slide and use them to find the height of the tree.

17. How tall is it? 3
Measure these 3
angle

18. How tall is it?
How do your results compare with others’ results? What are the sources of errors for each of the methods? Which method do you think is most affected by errors?

20. Teacher notes: Maths Outdoors
In this edition are a few short ideas for mathematics activities outdoors. Different ideas are suitable for different age/ attainment ranges from KS3 to KS4. Opportunities for problem solving and reasoning outdoors are abundant, and seeing things in different contexts and putting mathematics into practice can help students make connections as well as engaging them with the mathematics. Some of these activities could be done in a large room instead of outside.

21. Teacher notes: Magic Loops
Students should be familiar with Pythagoras’ theorem as the loops produce a 3:4:5 triangle. This activity can be carried out indoors using a chair or similar. The larger the object, the more accurate the result is likely to be.

22. Teacher notes: Place your book
Students should be familiar with Circle Theorems. You will probably need to look at the diagram to understand what is required. Encourage students to go on both sides of the posts. This is a good example of reasoning ‘in reverse’, i.e. the opposite to what they would normally expect within a question, which is a feature of the new GCSE examination questions. Students should know that the ‘angle in a semi-circle’ is 90°, however, in this instance there are a collection of 90° angles, subtended from the same points (diameter) which form a circle.

23. Teacher notes: How tall is it?
Three different methods are suggested. Methods 1 and 2 simply use similar triangles, method 3 uses trigonometric ratios. Spotting the similar triangles in a question is often a stumbling block for students, so memorable activities may help them to remember to look for them, particularly as questions are often placed in these sorts of contexts. One of the things to consider when thinking about how much difference an error of a single degree makes to the final answer

24. Teacher notes: How tall is it?
One of the things to consider when discussing how much difference an error of a single degree makes to the final answer is a tan(x) graph. For values up to about 50°, there is little difference from one degree to the next. From 70° onwards, the differences are much greater. This suggests that measurements taken a bit further away from the tree (giving a smaller angle) are likely to be more accurate.