Not that long ago I was talking to a true great in education research, Zalman Usiskin. When we were talking about length and perimeter he questioned why we used perimeter in the title for this particular subset of waypoints in the framework. Zalman felt that in fact we should use the term paths, as a perimeter is a special type of path. I’ll be honest – this stopped me in my tracks and really made me think about what length, paths, perimeters are all about.
Not long after this discussion I was discussing fundamental ideas in maths with Martin Hyland. We were thinking about what some of the basic concepts are that we may take for granted but that younger children need to develop explicitly. Very often these ideas are crucial for later mathematical progression. The specific example applicable here is the fact that the shortest path between two points is the straight line between them (assuming they lie on a flat plain) and this is normally defined as the distance between the two points.
My mind wanders in this role. It’s a mix of beautiful fractals (I’ve been looking at similarity recently), carefully constructed Euclidean proofs (which I’ve never really completed in their full complexity) and arguing voices regarding the best approach.
Most recently whilst reading about the Erlanger Programme, which emphasises transformational geometry over Euclidean, I thought about how geometric shapes relate to time-distance graphs. This was seeded by a comment that motions (in the mathematical sense) and movement or displacement are not the same thing; which is a whole other argument and blog!
Imagine walking along the path that is the perimeter of a shape at a steady rate. Your time distance graph would look something like this:
![](data:image/png;base64,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)
But, what if the time distance graph was with respect to the ‘centre’ (I’ll leave you to decide how you define that one!) of the shape?
In the simplest case, a circle…
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaYAAADyCAIAAACJe/YBAAASrklEQVR4nO3caVgV9R7A8dEsBdNyySVvWgi54I7mLcpEBRWyROgqgrlritt1V3BJTa/mgmiCpGWoqanlroB7pYKgGCiLC24ossi+nnP43xcgmqGCt+A+z+/7eeYFzMwZfufFfJ9zmDlHUwAghlbeAwBA2SF5AAQheQAEIXkABCF5AAQheQAEIXkABCF5AAQheQAEEZq8+/fvX7t2rbynAFDWhCZv3bp1VlZWGRkZ5T0IgDIlMXlnz57t1auXsbHxjBkzDAZDeY8DoOxITJ6bm9tLL72kaVq7du0iIiKoHiCHuOQdOXJk0KBBtWvXrl69+qBBg6ZMmZKQkFDeQwEoI7KSp9PpHB0dZ8+ePWzYMBMTk8DAQHNz8yNHjpT3XADKiKDk6fV6f39/CwuLvXv3rl27tmnTpkqpIUOGjB07NjQ0tLynA1AWBCUvJyfH0tJy4cKFmZmZnp6epqamSqmwsDALC4tVq1aV93QAyoKU5KWlpX3//fdt2rQJDAxUSi1btqxx48YFmyZPnjxixIhjx46V53wAykSpk5ef//gPf/jx/1VISEiHDh08PDxSUlKUUkuXLi1K3uXLlx0dHV1dXfPy8sp1RgB/uz8nz6DUzSDf9Wvnz5s3b+5Ds9zdNv36W6xBKaUKC3cneOOqH78/cObJFzzz85VSyRmZafdi/5bpSygxMdHb29vCwuLWrVsFax5NnlJq/vz5ffr0OXDgQDkNCKCM/Dl5eUodcq9v9LpWrYZp61YtzAu0bPaG8Tt9+y3aF30368FtbBe3DbedMPhL36d9cCsr/qLP3l0BB07/LdOXUExMzHfffbd+/fqiW/AeS15YWJiXl5evr285DQigjBSbvMNzGr7cd/TmPUmPrteHLu1hZfK22WfbrheseOSN7ZOPf8aj4+u9PhzvefmvGfev8ljyAAhRfPLmNnrZcfj6H289tilyv3svh9c7zghShj9U7inJC1ljZfJp9yleV58xRxn/O5DkATKVKnkq7TfPZS7v1vtsb4w+X6lL24b2GD9wvu9NpVRO6k6X91s2qlnrtbo1Xq1l0bvTugiljnq829RU0ypWqPCieY8OX51XSil1be/IT6xfefXVunXr1a1br6Fpg0l+wbF6pRIv7l04vcM/vww5Pde8ebuaNavXeLVpz+HuJ3VFfz399NKvHKpWr1GnXr0GtetO23ApIVMp9eBtdtSyVp1b1Kpes3aNV5u07eq+N+tpT5vkATKVLnnq9r5d0z/W/jH7iC5fqeA1H/zDznKMxzld5rHFdi1b9Zj41Zdem7ZuWTjD1a5tXcsVgTdP7Vz0uVnNNk279F1++KcL95SKPbnAskU7W7vBC7f8sOnbNcu/GdFWe731AK/zt1Xype2utppW32L6PM+16zZu3jJ/4JAe771ms/zofWVQ2VFbZ46wteriOHLp5k3frPUe2/619kNWrAtNVkplJ4YemNXLwmHInCUrfTdt9l0yfZ6TjUnPxftiDU/qHskDZCpl8uID/Gf2faHy+J90+Uqd+8bazMFmmvfp2/e8bDWtz4qi6xi3/LaO+feGC0qpyA02pk52bt/eKdhw75TH2Mnrj/6S8mDP5LW2Rtr7E/Yfzcy8untCL61KzQ5rfit8YRd+/D+OppXbzQxWuryg1d1MLDoOm3QorWBb6s/W5nUa20z66YJelxjoNajW29233kh+cNTcU552pnWsvEMjkov/ygCSB8hUyuTd8/eb41ypwSx/Xb5S53yszRy6TfY6n5a8b/JbVbpP+G7L9mOH/U9HRqYWPSBo1YdvOdhMWhX16FHiY8NO7Nt30C9g78GARc7Gmv10vxM5GdE7xv7rpTp2W1LScwp2izu/Zcwg4/ojDip9pEffRtZOU34+V3gEg8o66encdfSsrYcv37qxbXxjzWHhvv17jh32P3jA79ipvVt9FvbUNPvvjl9JK/ZpkzxAplIm78ae3YtcKn/ke0Wfr9S5tdZmDlbjPK8plRmysXett2oYVdI0rWG3dxacyEpLSctVSgV93fnR5OnzsjNvbB7p8m4NTatc1djY2Liypmkfzj5yKicjavsYZ6P6owNUauEdwbeDfUf1f7nhKD+Vf3RS+9f72i84HPPnZ5AUeH6ZtaZVfNHIyLiQkbGRUbXXGlb7fFfI7cxinzbJA2QqXfLiA+a59Wvw8sTf8gxKqRBvazMHq7ErIpRS+QZdnkGp1EPTRn/4iqZp1eqajDqoslWIT5ei5BmUurq1X/2GWs1/jvz2UKpSKj03d/eUGi9+OvXQ8ZyMqO1j+hvV/9xPpTyavKoNR/mp/ONT3mnQ135+wJ8v/OZfO/Cza9sKL7puvJeSrPLz8x6hNzzpQjDJA2QqcfLylVLJZ5b3G/BGkwEHCm5SCfa2NnOwGrMi8pHdcpJTkiIO+04f3baq5cqryerUWisT+26TVl9WSmWoa96f1m7r4vajf1x+4d9K2uD0SkW7qYdOPjF5b4zyV/orqwe81dVp4o6gh9Nc3GjfcdyEDXsCw35fZK1pH628aij2a92LrR7JA2R60q3IVfuN3XYw/Q8b7uxxdWnXpFnXRb9lF9yJF+xtbebQbdLXIenpAYttbNedeJCctKDFY1/WbL9PSVbXdzibDv1sxrd3lVJ6lbrBpeKrNhP3Hi+4kqqPOjDsjdqa1nn2iSBddtR212KSZ/yPUf4qV/f7OrtGLVs5Dd/x4NNtF8a9X9G0y8gfApUu8Zel9q9Vajj4p18fXL8wJPh59ewxemNUTAqXLwA8VGzy/NzrVW5l0qXnmAVfzHYrMGv2qEH9+vQdudjT/+aDO+UCPS3r93x39PJzhqxfPB3M37cd9++xs+d+MWf4yCG97Gym/ByldEpFrLQdZteu3VDv5TvOJanoEyPat+nQ873+Uxd8MXXmjKFjhvbrVENr4OztGxNzedsIe63qwP0qJbfg+DcD1w/urVUfdFClq7zYw1+79bexsbVznTvXbcaMz9/7oFt/D+/A2Cyl9BnRpzz7v9e0o+Mw18lz582ZM3rcqJ4ftO43e1fM3Yzi39mSPECmPydPr9S5LUP7O7Zr27Zd69ZtCrVo8naPubv9C+41KcxI1N6pn82d6LnzjlJKl+U3pneXd5qbt2zRtIml45g5ZwoPZwjzXTXOytTC1nqK302l1J1t7sPtzJubmzdr9n73T5aGZQS429tPXesbHnHvlNeXPT9dcVZlFDY1ISrA40vbvh7BKlOnlFI5od/4jDBr0qxVmzbtzFsu2HE9NU8V3Yp8/xc360/ea9m8ZeuWzZt17O406VjhUYptHskDZCr9l0c99aen7P7MXR/fM/+xX4t76PN+TI3kATJJ+YrQx5A8QCaSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQocn76qslJm+9Wd5TAChrQpO3fIVH47eb65XKyNals7CwlGrJysvI1uXpDOV9Hj8Pocn76ci5Vk5Le31x9KO5hz+aw8LCUoql56yAz5ae3HPmZnmfx89DaPJWrfExa/XujcTc6NjUqNssLCylW67eTUtOzy3v8/h5CE3ekiWLTd5sWN5TAChrQpPHFVtAJpIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB0AQkgdAEJIHQBCSB6BEDAZDRkZGfn5+eQ/yPyF5AErk/v37O3fujImJyc3NLe9Znh/JA1AicXFxkyZNatCggaura3R0dHmP85xIHoASyc3NjYyM/OGHH1xcXCwtLadMmRITE1PeQ5UayQNQOidOnHB3d7e3t3dycvLx8UlKSirviUpBaPKWLFlC8oD/hb+//5AhQxwdHRctWnTixInU1NTynqhEhCZv5cqVjRo1unLlSjiA0rt06dKNGzdCQ0MnTJjwwgsvtG/ffs2aNVevXi3myoYuJyv+ysWL4WFh4eHh4RcvRURERly6GB4eHh4e9vuFyFt3UtKzs+Jvh1+KTcvX/f1Xg4Um7+DBg82aNdM0TdO0CgCeV8WKFStUqFBwKtnb24eFhT12rhniw8/Oa1XlxYqapj3YUdO0ClrBw+o5j9q8/9yZBQNfqD5sd15c9t9+7gtNXm5ubmJi4l0AzyshISExMTE4OPjjjz+uU6fOiBEjzp8/X8zJZtDlpSfE3b17525Cas6ZH6fb9qrVbMTutFuxd+/F3Ym9l5yalZOXl55yNy4lRxl4lQfg/9OdO3cWLVrUuXPncePG7dmz59atW3q9/lkPun1yyb+c6radGvQXDPB890STPAClc/36dQ8PD2dn5wEDBsyfP//06dMl/kjGtcMLHf5Vt/XEX5XOULQyPT7qwA+TZ24LN+SoO0E+S7Zt3rQ78Jdv+zsPGTx4wMDB7ltDr6SmZIZtnu3kMnjw4AEDh8zaevZC4sODJge4LZzk4uTy2YABo8fN8A3Oe9oEJA9AiRgMhvj4eD8/v2nTpnXv3t3Z2XnXrl2lPEaxybt7af/EPprWb7suRZ37pksjy3pturn+Z2LXLjbWNj2aV6vfdZjTpK+8pw3r1aWrtbVNj+ZVa1uOGb85OlOpfF3S1aMbZw7u8JHNP7va9LB+19K6XddPFh8NSdTrnjAByQNQIqmpqb6+vh07duzUqdPGjRtzcnJKf4xik3cvyt9twAvVh+7WpaoLW/q2NdHebGG/ObJg422vwR2raZpJ+34/XylYc2uV49vNW/VZvi9L5dza7W5aocaYPb8WvuhLSt8zvk4l0wHbL0dnFT8ByQNQIjdu3Bg5cuTq1auTk5Of9xjPTp5vr2YftHQaG1B07TbE29rs027jPS8X7R/s1cXE0Wba2ovpKce/6KgN33A57p5ShtxcnVLpcdH7hmsVnXx+DkspdgKSB6BE9Hp9Wlpadvb/ciPJs5O3wa6JfadRSy8WbQ327mbm2GXcyqiiNWfXdDFx6DFz3dmEpM19K2mVKr1UuYqRkVGVKkZGxlVeqlRJ0zTbtXvCSR6Acvbs5H1n28Sxs6tHZNHWYK9uZg5dxj2y5uzXVo0/tZvleSw6ZO47mtbr35t27jp59Ih/ocNHTxy5GHc/s/jLxyQPQJkpUfIcPhy9IqJoa/HJc7R1X3PySsSiDzSt79c38ovNW7F3+ZE8AGXmL0ueQ/fp34RlpR12a/Vyva4LTgYVvYvVXw9a7/NTYOzdjOLvmyF5AMrM1YD5vR1qtpjwy6PJi4s8NKO/ZjR4ly5VhX7bvXHv9z9fdqlo69mvrd7q3cl1+cMIBq3q1LB35wmrYpTKCt0x8M06b/efun5/UHR0RMTx4/um936xVtf5vwXHGVRxSB6AMhNzdLGTS8N3pp36w6u86MNzhlR97fN9ujT1u2/vlv2sx3s+fE0X4mNr3q/HxNUPL18Er+3erJ/tlDWF31J6c4dLI/NqmqZV1DStcq1Gnb1vJj/5EgvJA1Bm9HlZaWlJyRl/uFHYoM/LTE9MSs9V+UqXk3o/LSUj6+E/53TZqffTUjOyn7gmPzct6X5C/L34+Pj4ewmJSSnZT/ssCMkDUFZK8anYP+9akjXPRvIACELyAAhC8gAIQvIACELyAAhC8gAIQvIACELyAAhC8gAIQvIACELyAAjyX0xklw2uWD+SAAAAAElFTkSuQmCC)
Why does this shape form this graph?
What about a square with side length m?
![](data:image/png;base64,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)
Why does this shape form this graph? What position on the square do the peaks and troughs represent? Now this is just my sketch, but what is the nature of the curves? Are they sections of quadratics?
What about a rectangle with side lengths n and m (n<m)?
![](data:image/png;base64,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)
Again why does this shape form this graph? What position on the rectangle do the peaks and troughs represent? As before, this is just my sketch, but what is the nature of the curves? Are they sections of quadratics? Are each of the curves translations of each other? Why are the values ½n, ½m and √(1/2n)2+(1/2m)2 significant?
In my mind this kind of activity is fascinating and would lead beautifully into looking at individual components such as the height above the centre of a shape. Consider this in relation to the centre of a circle and experience another beautiful entrance into the fascinating world of trigonometry.
Another approach would be that of Texas A&M Society of Physics students. Rather than travelling around the perimeter of the shape they designed a floor that would allow you to ride a bike with square wheels without bumping along. (https://www.youtube.com/watch?v=LgbWu8zJubo) Same problem different situation, still captivating, and now I want to ride a bike with rectangular wheels.
SOMETHING TO TRY:
KS1: Having marked a spot in the classroom ask the pupils all to stand exactly three paces away from the spot. What shape have they made? What’s special about this shape? How could they use string, a drawing pin and a pencil to draw one?
KS2: Choose a selection of shapes each with a point identified somewhere on their interior. Ask pupils to find the places on the shape’s perimeters that are the furthest away from the point and those that are closest. Can there be more than one place? Ask them to explain their reason/method.
KS3: The red dot travels around the square at a constant speed. A graph is drawn of the time (x-axis) against height (y-axis) from the shown starting position. The two possible versions are shown below. Why are there two? What’s happening in each? What are the significant features of the graphs?
![](data:image/png;base64,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)
Draw the equivalent graphs for the following rectangle:
![](data:image/png;base64,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)
KS4: Draw the time distance graph, with respect to the centre, as you travel around a regular hexagon at a constant rate. Identify key positions along this journey on your graph.
KS5: Explain how you would draw the time distance graph, with respect to the centre, as you travel around a regular n-sided polygon at a constant rate. Identify key positions along this journey on your graph.
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