Paper folding and Proof

Square Folds For each of these challenges, start with a square of paper. Fold the square so that you have a square that has a quarter of the original area. How do you know this is a quarter?

Square Folds Fold the square so that you have a triangle that is a quarter of the original area. How do you know this is a quarter? Can you find a completely different triangle which has a quarter of the area of the original square?

Square Folds Fold to obtain a square which has half the original area. How do you know it is half the area? Can you find another way to do it?

Isosceles Triangle Start with a piece of A4 paper. Fold as shown edge to edge.

Isosceles Triangle Fold as shown, again edge to edge.

Isosceles Triangle Turn the shape over. It looks like an isosceles triangle. Prove that it is isosceles.

Kite (1) Start with a piece of A4 paper. Fold as shown edge to edge.

Kite (1) Fold as shown.

Kite (1) Turn it over. Prove that this is a kite.

Start with a piece of A4 paper. Fold as shown. Equilateral Triangle Start with a piece of A4 paper. Fold as shown.

Equilateral Triangle Pick up the bottom left hand vertex and fold so that the vertex touches the centre line and the fold being made goes through the top left vertex.

Equilateral Triangle Pick up the bottom right hand vertex and fold so that the fold being made lines up with the edge of the paper already folded over.

Equilateral Triangle Finally, fold over the top as shown. Folding away from you helps keep the triangle together.

Equilateral Triangle This looks like an equilateral triangle, but how can we be sure that it is? Unfolding the shape gives us lots of lines to work with, but all we really need is the first couple of folds. It may help to start with a fresh piece of paper and simply make these two folds.

Equilateral Triangle Drawing round these edges will also be helpful. This is what you should have. What can you work out?

Kite (2) Start with A4 sized paper. Fold as shown below.

Kite (2) Fold corner to corner, as shown.

Kite (2) Fold corner to corner, as shown.

Kite (2) This is a kite – or is it?

Start with a sheet of A4 paper. Fold corner C up to corner A. ‘Regular’ Pentagon Start with a sheet of A4 paper. Fold corner C up to corner A.

Lay BE along DF to create a mirror line. Mark the crease and open back out.

Fold BE to lie on the crease. Fold DF to lie on the crease. This looks regular but isn’t. Prove it.

Hint 1 The top angle is made up of two smaller angles. Find out what these are and add them together.

Haga’s Theorem You can easily find half way along one side of a square by folding. Can you find a third of the way along a side by folding?

Find the length of the side AE. Now find the length of DF. Does this help?

Why not use the shapes in KS3 and then introduce the proofs in KS4?

Instructions for the shapes can all be found at this link Paper folding instructions

Equilateral Triangle Hints

Equilateral Triangle Hint #1 The triangle is created as shown; we need to show that 2 of its angles are 60°

Hint #2 Can you find the angles in this triangle? Equilateral Triangle Hint #2 Can you find the angles in this triangle?

Hint #3 Think of the length of the short edge as 2x. Equilateral Triangle Hint #3 Think of the length of the short edge as 2x.

Hint #4 When you fold the vertex in, what distance is shown? Equilateral Triangle Hint #4 When you fold the vertex in, what distance is shown?

Hint #5 What dimensions of this triangle do you know? Equilateral Triangle Hint #5 What dimensions of this triangle do you know?

Hint #6 The fold line is a line of symmetry of the grey shape. Equilateral Triangle Hint #6 The fold line is a line of symmetry of the grey shape.

Teacher Notes

Teacher notes: Paper Folding Here we have activities which link paper folding and proof. These support the development of reasoning, justification and proof: a renewed priority within the new National Curriculum. They are presented in approximate order of difficulty. Discussion and collaboration are key in helping to develop students’ skills in communicating mathematics and engaging with others’ thinking, so paired or small group work is recommended for these activities. The first ‘Square Folds’ is suitable for many ages and abilities since a range of responses with different levels of sophistication are possible.

Teacher notes: Paper Folding The next activities require students to create specific common shapes and then prove whether or not they are the shapes they appear to be. The first and second require a knowledge of Pythagoras’ theorem and surds, the third requires basic understanding of trigonometry including exact values. Hints are given for the equilateral triangle, but it is helpful to allow students to grapple with the problems by not showing these too soon. Once a hint is given, it’s useful to wait a while before showing another. Discussion and explanation are key. One option is to print the hint slides out on card (2 or 4 to a sheet) and just give them to students as and when they need them rather than showing them to the whole class.

Teacher notes: Square Folds There are a range of possible responses for each of these. One of the key concepts that can be developed through this activity is an appreciation of what is meant by proof. The progression through: Convince yourself Convince a friend Convince your teacher is a good structure to use to encourage students to think about ‘why’ something is as they think it is, rather than just responding with ‘you can see that it is’. Demonstrating by folding to show that different sections are equal may be acceptable, or you may wish students to consider the dimensions of the shapes.

Teacher notes: Square Folds Some possible answers: A square, a quarter of the area. The lengths of the sides are ½ the original. 𝑥 2 × 𝑥 2 = 𝑥 2 4 2&3. A triangle, a quarter of the area. 1 2 𝑥 2 ×𝑥 = 𝑥 2 4

Teacher notes: Square Folds 4&5 A square, half the area. Folding the vertices of the original square into the centre is one way to demonstrate that it is half the original area. Using Pythagoras theorem to show that the new square has side length √2 is also possible. This one is harder to justify. The diagonal of the new square is x. Using Pythagoras’ theorem, the side length of the square must be 𝑥 2 so the area is 𝑥 2 2

Teacher notes: Isosceles Triangle To show that BEC is isosceles we need to show that BE = BC. Using the fact that the ratio of the sides of ‘A’ paper is 1:√2 we can say that AB is of length 1 and the length of BC is √2. Because of how we did the first fold the length of AE is also 1 so using Pythagoras’ theorem in ABE we find that BE is also √2. So BE=BC and BEC is isosceles. A B C D E

Teacher notes: Kite (1) To show that BDEG is a kite we can show that BG=BD and DE=EG. Again we can say that AB is of length 1 and the length of BD is √2. Using Pythagoras’ theorem in ABG we can deduce that BG is √2. So BG=BD. FG=√2-1, as does FE. So DE = 1-(√2-1) =2-√2. Using Pythagoras’ theorem EG² =(√2-1)² x2=6-4√2. So EG=√(6-4√2) = 2-√2. So BG=BD and DE=EG and therefore BDEG is a kite. A B C D E F G

Teacher notes: Equilateral Triangle Making the equilateral triangle is relatively straight-forward and can be used with all students during work with shape. Do they tessellate? Fold the vertices in to the centre to make a regular hexagon etc. The more challenging aspect of this activity is proving that it is indeed an equilateral triangle. Assuming the length of the blue side is 2x. The red line aligns with the blue side when the fold is made, so it is also 2x. In the right angled triangle shown, Opp is x and Hyp is 2x, therefore the angle shown is 30°

Teacher notes: Equilateral Triangle Making the equilateral triangle is relatively straight-forward and can be used with all students during work with shape. Do they tessellate. Fold the vertices in to the centre to make a regular hexagon etc. The more challenging aspect of this activity is proving that it is indeed an equilateral triangle. Assuming the length of the blue side is 2x. The red line aligns with the blue side when the fold is made, so it is also 2x. In the right angled triangle shown, Opp is x and Hyp is 2x, therefore the angle shown is 30° and the other angle in the triangle must be 60°

Teacher notes: Equilateral Triangle Looking at the grey kite, the fold line is a line of symmetry. This gives the angles shown. Since the angle sum of a quadrilateral is 360° the missing two angles must each be 60°.

Acknowledgements Thanks to Carol Knights for the basics of this PPT which appeared in MEI’s Monthly Maths in February 2015 Ideas for ‘Square Folds’ from http://youcubed.stanford.edu/task/paper-folding-fun/ Images from: http://kimscrane.com/images/EF21K3.jpg http://www.muji.eu/images/products/l/4547315453153_l.jpg http://fc00.deviantart.net/fs71/i/2010/257/0/b/origami_paper_pattern_by_tseon-d2ypu5c.jpg

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