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To develop some generalised ‘rules’ and relationships, including the finding and expressing of a two-part rule.
Recognising the importance of examples and non-examples in reaching a generalisation.
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| Resources |
Vocabulary |
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Isometric dotty paper
Dotty grid on board
Scissors
Roofs notesheet – one each
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equal spacing, straight sides,clockwise, closed shape, trapezium (roof), rhombus(diamond), parallelogram |
| Organisation |
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Near ability pairs in mixed ability groups of 4 to 6
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| Whole Class Preparation: (about 15 mins) |
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Display a large sheet of isometric dotty paper. Make sure it is oriented like the notesheet.
Distribute the isometric dotted paper and the notesheet
What do you notice about this paper? Triangles, equal spacing.
Make as many different, four-sided shapes as you can. Ensure the rows of dots on the paper are horizontal. Lines must be drawn to the nearest dot in any direction.
Discuss similarities and differences. Some shapes will be the same but differently orientated. Some will be larger or stretched versions of others.
Focus on the roof shape and talk about the significance of the lengths of sides.
We can draw a roof from a set of four numbers; some sets of numbers make a roof, and some do not. Give children some examples, including the two already drawn on the notesheet. Emphasise that the numbers refer to the length of the line, rather than the number of dots.
Ask the children to use the convention of starting at the bottom left corner, then drawing up to the right, across the top to the right, angling down to the bottom right, and finally across the bottom to the left.
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| Group Work: (about 5-10 mins) |
Give out the Roofs notesheet.
In pairs, find out which sets of four numbers make a roof and which do not.
Write down any patterns that you notice and use this information to create a rule.
If necessary, change the rule as you work.
Ensure that children are drawing roofs correctly, starting at the bottom left corner and working clockwise.
Introduce counterexamples to children who have discovered part of the rule (such as first and third numbers the same: 1, 6, 1, 1).
Identify children who have noticed rules at different levels to contribute later.
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| Class Sharing: (about 15 mins) |
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Ask the children you identified earlier to contribute their findings, beginning with the simplest ideas. List rules on the board, valuing all patterns involving odd and even numbers, last number always the biggest, and so on.
Go through rules one by one – Are these true? Does this always produce a roof? Are there any ‘not roofs’ produced by this rule?
Discuss rules that are similar and how they are similar.
Agree that some rules only apply in a few cases but not all.
Make up sets of numbers with the children – What could we put in the 1st place? What about the 4th place? Try some big numbers. Let’s start at the end.
What if we used n (any number) and m, or ? Last gives , , , +
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| Paired Work: (about 10 mins) |
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Supposing we only have the length of the base of a roof. How many roofs could we draw on top? Model an example on the board for base of 4. There are three possible roofs (tell them the triangle is not allowed).
Try with another base length, such as a base of 5 or 8 or your own choice.
Write down anything you notice.
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| Class Sharing: (about 5 mins) |
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Choose several children to share examples and rules.
What if the length of the base was a big number, such as 100 or 350?
What if the length of the base was n?
What did you do first in this lesson and how did you proceed? Constructed examples. Then what did you do with the examples? Looked for rules and then expressed the rules in words or symbols.
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