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									           The Garden Path Problem
         An 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
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
                   The Garden Path problem

                                7m

                                5m



  5m    3m




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?
Teachers guide

Students should get the answer 20cm2 and see that this is
equivalent to 3+5+5+7.

They can then try this for other numerical solutions but should
then move into algebra as per next slide
                   The Garden Path problem

                                ?m

                                Am



  ?m Bm




What will be the lengths represented by the question marks?
                         A+2


                            A

           B
 B+2




Can you now prove your numerical generalisation?
PROOF

AREA = (A+2)(B+2) – AB

= AB+2A+2B+4-AB

= 2A +2B + 4

ADDING SIDES GIVES A+B+A+2+B+2

= 2A+ 2B + 4

THEREFORE ADDING SIDES WILL ALWAYS GIVE THE AREA

								
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