The Garden Path Problem An approach to problem solving

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The Garden Path ProblemAn approach to problem
solving

• This is an example of a problem or starting
  point that enables pupils to explore and
  investigate new mathematics. It enables them to
  solve increasingly demanding problems and
  evaluate solutions explore connections in
  mathematics across a range of contexts generate
  fuller solutions
• Represent problems and synthesise information in
  algebraic, geometric or graphical form move from
  one form to another to gain a different
  perspective on the problem

8.1
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Using and applying mathematics
Objectives addressed in Garden path

• Suggest extensions to problems, conjecture and
  generalise identify exceptional cases or
  counter-examples, explaining why justify
  generalisations, arguments or solutions pose
  extra constraints and investigate whether
  particular cases can be generalised further

8.2b
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The Garden Path problem
7m
5m
3m
5m
A metre wide path surrounds a garden.The area of
the path is ? What do you notice about the
dimensions and the area? Will this always be
true?How can you prove this?
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Teachers guide Students should get the answer
20cm2 and see that this is equivalent to
3557. They can then try this for other
numerical solutions but should then move into
algebra as per next slide
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The Garden Path problem
? m
A m
B m
? m

What will be the lengths represented by the
question marks?
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A2
A
B
B2
Can you now prove your numerical generalisation?
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PROOF
AREA (A2)(B2) AB AB2A2B4-AB 2A 2B
4 ADDING SIDES GIVES ABA2B2 2A 2B 4
THEREFORE ADDING SIDES WILL ALWAYS GIVE THE AREA