Name
Proofs that Square Root of 2 is irrational
Topic
Algebra
Learning time
280 minutes
Designed time
280 minutes
Number of students
25
Description
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?
Outcomes
  • Reproduce Reproduction proofs containing sophisticated skills. Reproduction proofs containing key ideas.
  • Investigate Investigate the width of proof after the Gardner's proof.
No outcomes are set
Editor
zenonlig
Read some characteristic proofs - creation of the working groups
80 minutes)
  • Read
    40
    25
    1
    Read the activity in SPARK Adobe text.
  • Discuss
    30
    25
    2
    Attention in proof no 1 and the proof 8 of "Laczkovich & Gardner" in "Cut the knot" .
  • Produce
    10
    25
    1
    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
    30
    25
    1
    The two proposed activities is the "spring from nowhere", for Tennenbaums's solution and the proof of Gardner
  • Collaborate
    20
    25
    0
    Find the similarities between the proofs of the two activities and the proofs of Tennenbaums and Gardner
  • Discuss
    10
    25
    0
    Write both proofs (Tennenbaums and Gardner) and find the "deus ex machina"
Notes:
Resources attached: 0
assessment
30 minutes)
  • Produce
    30
    25
    0
    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
    30
    25
    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
    40
    25
    0
    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
    40
    25
    0
    Solve the problems of activity and presented the work in Power Point.
Notes:
Resources attached: 0