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?

Pythagoras Unfolded Start with a square of paper. Put a dot a short distance from one vertex - about a third of the side length works well, not too close as you’re going to have to fold from it. Put dots exactly the same distance from the other vertices, working around the square as shown.

Fold to join the dots as shown. What shape is this? How do you know? Pythagoras Unfolded Fold to join the dots as shown. What shape is this? How do you know?

Pythagoras Unfolded Mark three other dots as shown, using the same distance as you used for the previous ones.

Pythagoras Unfolded Create folds as shown. Notice that two of the folds don’t go all the way across. It’s easiest if you do the vertical one first as this gives the guide line for where to finish the other folds.

Pythagoras Unfolded For the next slides, locate the relevant lengths or shapes on your folded square.

Pythagoras Unfolded a

Pythagoras Unfolded b

Pythagoras Unfolded c

What are the dimensions of this triangle? Pythagoras Unfolded What are the dimensions of this triangle?

Pythagoras Unfolded What is this area? c2

Pythagoras Unfolded What is this area? a2+b2

Pythagoras Unfolded Explain what this sequence of diagrams shows: Thinking about how you folded the shapes, can you explain why the yellow and green triangles will fit as shown?

Pythagoras Unfolded Can you now link it all together and explain how this is a proof of Pythagoras’ Theorem? This proof started with a square in which you created triangles in a certain way; would it work for any right-angled triangle?

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?

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.

Regular Pentagon Start with a piece of A4 paper. Put two diagonally opposite vertices together and fold as shown.

Turn the shape clockwise so that it looks like this: Regular Pentagon Turn the shape clockwise so that it looks like this:

Crease along the line of symmetry, then open it back out. Regular Pentagon Crease along the line of symmetry, then open it back out.

Fold so that the red edge meets and aligns with the red fold. Regular Pentagon Fold so that the red edge meets and aligns with the red fold.

Fold the other side (marked in red) in a similar way. Regular Pentagon Fold the other side (marked in red) in a similar way.

Regular Pentagon This looks like a regular pentagon, but how can we be sure that it is? Folding can be inaccurate, but with mathematics we can prove whether it is regular or not.

Regular Pentagon Unfolding the shape gives us lots of lines to work with, but all we really need is the first fold. It may help to start with a fresh piece of paper and simply make the first fold.

Regular Pentagon You need to know that the dimensions of a piece of A4 paper are 297mm by 210mm. What is the size of the internal angle of a regular pentagon? Can you work out the size of the angle at the top of the shape?

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

Hint #2 Draw this line on before opening it out Regular Pentagon Hint #2 Draw this line on before opening it out

Regular Pentagon Hint #3

Hint #4 Why are these the same length? Regular Pentagon Hint #4 Why are these the same length?

Teacher notes: Paper Folding This month’s edition has 4 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. The second activity looks at a paper-folding proof of Pythagoras.

Teacher notes: Paper Folding The third and fourth activities require students to create specific common shapes and then prove whether or not they are regular. The third requires a knowledge of basic trigonometry, the fourth requires basic understanding of trigonometry and Pythagoras and the ability to expand brackets and solve linear equations (a quadratic term appears, but is balanced by an equivalent one on the other side of the equation). Hints are given, 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 explanaiton 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: Pythagoras Unfolded Although it would be possible to skip straight to the diagrams, students having something they’ve folded in front of them helps them to see and understand exactly what’s going on. They can then annotate and/or colour the square and stick in their books or folders. Identify that the bottom triangle is a,b,c. Show that a2+b2=c2 The proof will work for any right-angled triangle. Create 4 copies of the required triangle and arrange them so that a ‘long’ and ‘short’ side form the edge of a square as shown (makes no difference which is short and which is long for isosceles triangles).

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°.

Teacher notes: Regular Pentagon The ‘top’ angle consists of a right angle and the angle shown. Since the interior angle of a pentagon is 108°, if the marked angle is 18° then it might be a regular pentagon, if it’s not 18° then it definitely isn’t regular.

Teacher notes: Regular Pentagon The red line is created when the fold is made.

Teacher notes: Regular Pentagon The dimensions of the triangle are as shown. Using Pythagoras’ theorem: (297-x)2 = 2102+x2 2972-594x+x2=x2+2102 594x =2972-2102 594x = 44109 x= 74.26

Teacher notes: Regular Pentagon The dimensions of the triangle are as shown. Using trigonometry: tan 𝜃= 74.26 210 𝜃=19.47° This means that the top angle is 109.47°, so the pentagon is not regular… despite the title of the activity!

Acknowledgements 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 Pythagoras activity from T. Sundara Row (1905) Geometric Exercises in Paper Folding.