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Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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What do these two triangles have in common? How are they related?

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Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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How many different triangles can you make on a circular pegboard that has nine pegs?

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Make a flower design using the same shape made out of different sizes of paper.

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Can you find all the different triangles on these peg boards, and find their angles?

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Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

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Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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I cut this square into two different shapes. What can you say about the relationship between them?

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Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?

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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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This activity focuses on similarities and differences between shapes.

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Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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In this game, you turn over two cards and try to draw a triangle which has both properties.

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The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?

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Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

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You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

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Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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Can you each work out what shape you have part of on your card? What will the rest of it look like?

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Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

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How would you move the bands on the pegboard to alter these shapes?

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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form?

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Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

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A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

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A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

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Make an equilateral triangle by folding paper and use it to make patterns of your own.

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Start with a triangle. Can you cut it up to make a rectangle?

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Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

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If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

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ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?