The activity looks at the meaning of concepts of "key ideas" and "memorablity" and how they relate to the metric "width of a proof". It attempts to show whether and how they are congruent with other aspects of proof discussed in literature on the teaching of proof and proving.
We presented 4 differents proofs of the irrationality of SQRT(2). Three proofs utilise algebraic tools, and one proof utilise geometrical tools.
Aims
1) To what degree does the width of a proof (as Gowers uses of term) represent a new idea in mathematics education.
2) How does memorability (as Gowers uses the terms) relate to understanding, and how could the concept be of benefit to mathematics education?
Read some characteristic proofs - creation of the working groups
80 minutes)
Read
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Read the activity in SPARK Adobe text.
Discuss
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2
Attention in proof no 1 and the proof 8 of "Laczkovich & Gardner" in "Cut the knot" .
Produce
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Students are divided into groups of 2 and carry out their first job. Write the proofs 1 and 8.
(reproduction in PADLET step by step the profs 1 and 8)
Notes:
Resources attached: 0
Two activities that support the width of proofs, according to Gowers
60 minutes)
Investigate
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The two proposed activities is the "spring from nowhere", for Tennenbaums's solution and the proof of Gardner
Collaborate
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Find the similarities between the proofs of the two activities and the proofs of Tennenbaums and Gardner
Discuss
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Write both proofs (Tennenbaums and Gardner) and find the "deus ex machina"
Notes:
Resources attached: 0
assessment
30 minutes)
Produce
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Show that the sqrt (2) is irrational number, using an proofs that you think you can replay with the most complete way.
Notes: Investigation of Cowers proof
Resources attached: 0
irrational Number and continued fractions
110 minutes)
Investigate
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2
Read in wikipedia, the paragraphs:
1) Calculating continued fraction representations
2) Finite continued fractions
3) Infinite continued fractions
4) Generalized continued fraction for square roots
Can you help the investication, by the activity abour SQRT{2} in padlet
Collaborate
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1) Define the collaborative project.
2) Identify project elements and components in detail;
3) For each component identify the resources that are essential. These can be;
a. materials
b. equipment
c. strategies
d. knowledge
e. experience
Produce
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Solve the problems of activity and presented the work in Power Point.