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Making Money The Royal Mint has announced the introduction of a new £1 coin for 2017. It will be based on an old threepenny bit – a 12-sided coin in circulation from 1937 to 1971.

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Making Money Many coins are made by cutting out blank shapes from a long strip of metal and then stamping the appropriate design on them. In the UK we currently have circular coins, coins which are heptagonal (50p and 20p) and in 2017 we will be getting a coin which is a dodecagon. Which of these shapes do you think are most cost effective to make?

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Making Money If circular coins are cut from a strip of metal: How much is wasted? How can wastage be minimised? What information do you need to know?

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Making Money 5p coins have a diameter of 18.0mm If there are no gaps between coins or between coins and the edge of the strip, how much metal is wasted producing the coins?

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Making Money: Hint Work out the wastage for 1 circle enclosed by a square.

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Making Money If coins were arranged on very large sheets (partially shown) how much wastage would there be?

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Making Money: Hint Determine what type of triangle this is and then find its area.

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Making Money: Hint How many of the white pieces are there for each circle?

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Making Money: Exploring Further If heptagons or dodecagons are cut from a strip of metal, how should they be arranged to minimise wastage?

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Teacher notes: Making Money This activity involves problem solving, finding areas of plane shapes and using Pythagoras theorem or trigonometry. The activity could also be approached in practical ways which do not require the use of trigonometry or Pythagoras theorem.

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Teacher notes: Making Money Slide 4 Information that might be needed: How long the strip of metal is How big the coin is How close the coins can be to each other as they are cut out of the strip If the coins can be cut out right to the edges of the strip

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Teacher notes: Making Money Slides 5 & 6 Area of a circle within a square Area of square = 18 x 18 = 324 mm 2 Area of circle = π x 9 2 = 254.5mm 2 Wastage = x 100 = 21.5% This is replicated along the strip of metal. 324 – 254.5 324

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Teacher notes: Making Money Slides 7, 8 & 9 Each side of the triangle is formed from 2 radii, therefore it is equilateral. Height of the triangle (using Pythagoras theorem) h = (18 2 - 9 2 ) h = 243 h = 15.59mm Area of triangle = ½ x 18 x 15.59 = 140.31mm 2

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Teacher notes: Making Money Slides 7, 8 & 9 Each side of the triangle is formed from 2 radii, therefore it is equilateral. Each of the segments is 1 / 6 of a circle White piece = area of triangle – area of half a circle Area of white piece = 140.31 – 127.23 = 13.08mm 2

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Teacher notes: Making Money Slides 7,8 & 9 Area of two white pieces = 13.08 x 2 = 26.16mm 2 Wastage = 100 x (area of two white pieces ) ÷ (area of two white pieces + area of a circle) Wastage = Wastage = 9.31% However this is for an infinite sheet, in reality there would be additional wastage at the edges. The bigger the sheet, the less the relative wastage. 100 x 26.16 26.16 + 254.5

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Teacher notes: Making Money Slide 10 Pupils could explore fitting coins on strips of paper, deciding whether rotating coins or having them all the same orientation will fit onto a shorter strip. They could do this with real coins (or cut-outs!) or by using a dynamic geometry package.

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Acknowledgements http://www.bbc.co.uk/news/uk-politics-26632863http://www.bbc.co.uk/news/uk-politics-26632863 accessed 31/3/14 http://www.royalmint.com/aboutus/policies-and-guidelines/advertising-guidelines accessed 31/3/14 http://www.fleur-de-coin.com/articles/unusual-coinshttp://www.fleur-de-coin.com/articles/unusual-coins accessed 31/3/14 http://www.telegraph.co.uk/finance/currency/8655340/How-the-UKs-coins-are-made-and- what-they-are-made-of.htmlhttp://www.telegraph.co.uk/finance/currency/8655340/How-the-UKs-coins-are-made-and- what-they-are-made-of.html accessed 31/3/14

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