# Magic Loops

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## Magic Loops

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Slide1

Slide2

Magic Loops

You will need:A long length of stringA tree, large fence post, chair or other objectSome friends – groups of 4 work best, but 3 would do.

Slide3

Magic Loops

Tie

a small loop in the end of 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

again tie a small loop

.

Wrap your string a further 3 times and tie a final small loop. Cut off the remaining

string.

Slide4

Magic Loops

Unravel

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?

Slide5

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 knotsto check that walls are perpendicular.

Slide6

Slide7

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.

Slide8

Place the two

rounders

posts (students standing still) about 8- 10 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.

See diagram on next slide!

Slide9

Book

Look along the edges of your book to line it up

P

osts

Slide10

There should be lots of places to stand.

Once everyone is in place, what do you notice?

Why has this happened?

Slide11

Slide12

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.

Slide13

How tall is it? 1

You need:A metre rule and a long tape measureIf 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?

Slide14

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.

Slide15

How tall is it? 2

Measure these 3

Slide16

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.

Slide17

How tall is it? 3

Measure these 3

angle

Slide18

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?

Slide19

Slide20

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.

Slide21

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.

Slide22

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.

Slide23

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

Slide24

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.