Maria Luisa Spreafico – [email protected]
Dipartimento di Scienze Matematiche - Politecnico di Torino
joint work with Emma Frigerio- [email protected]
Dipartimento di Matematica - Università degli Studi di Milano
http://lnx.origami-cdo.it/friburgo2015/
5-year long project in an Italian primary school
3 – 4 workshops per year on origami and math,
from 1st to 5th grade (children 6-10 years old)
AIM: promote better understanding
develop critical sense
improve communication skills
help people afraid of math
increase long-term retention
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Didactics and Research of Folding – Freiburg 2015
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Start from the Real World
Didactics and Research of Folding – Freiburg 2015
Miri Golan and Origametria
Rules for students
•Watch and listen before folding •Put the model on the desk •Raise hand to ask/answer a question
Rules for the instructor
•Don’t tell what we are folding •Don’t touch students’ model •Make sure everyone has made the fold correctly
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The Van Hiele Model
1. Visualization
2. Analysis
3. Informal Deduction
4. Formal Deduction
5. Rigor
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Every child folds the mistery model, while working on a specific mathematical subject (diagonal, lines, angles, area…)
Or…….
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… many children cooperate together in a big
(modular) project
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First and Second Grades
Identifications of figures:
Easy problems: how many triangles do
you see in the model?
1 2
5
3 4
6
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1 2
5
3 4
6
There are
6+ (1,2,3,4,5,6)
2+ (1 3, 2 4)
2+ (1 2, 3 4)
2+ (1 3 5, 2 4 6)
1= (all 6)
____
13 triangles
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Description of figures,
their classification with respect to some property
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Third and Fourth Grades
Ex
Examples: 1) mutual position of lines and angles Benefits and limits of paper: abstraction is needed!
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2) Classification of triangles
(from an idea of Stefania Serre)
What if you fold using
•other right triangles? •acute or obtuse triangles?
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3) Fractions (and equivalent fractions)
EQUIVALENT FRACTIONS:
C = 14/15 B = 1/15 or
C = 28/30 B = 2/30
MOREOVER:
C+B= 14/15 + 1/15 = 1
Didactics and Research of Folding – Freiburg 2015 Pat
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Fifth Grade Workshop 1: we need a unit • Motivation 5 equal square sheets folded in different ways. Which shape has the biggest/smallest area?
• The mystery model: waterbomb cube
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Workshop 2: using different units
While folding a simple model (a traditional
box), at many steps we compute the areas of
white and colored parts choosing different
units.
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(a) (b)
In (a): W = 2 T; C = 8 T and C/W=4
(whatever unit we use)
In (b): W = 2 (t+S) = C
(decomposition)
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Final remarks: using the paper models, compare in the two boxes
Segments (2:1)
areas (4:1)
volumes (8:1)
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Workshop 3: an optimization problem.
First we fold a traditional envelope.
Then children work in groups of 4.
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1) Children color the outline of the closed
envelope and of the closing triangle CT, then
they reopen the model.
2) Children observe the CP: from the red line
they can reconstruct the starting sheet.
Didactics and Research of Folding – Freiburg 2015
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Suppose the sheet is a 16 x 16 cm square.
How long are the sides of the envelope?
PROBLEM
1) From which sheet do we have to start if we
want an envelope with the CT attached to its
short side?
2) Which sheet is optimal to economize paper?
Answer:
6 cm and 8 cm
Didactics and Research of Folding – Freiburg 2015
Children reconstruct the sheet needed for the second
situation and compare.
Area:16x16 cm² = 256 cm² 12x19 cm² = 228 cm²
Didactics and Research of Folding – Freiburg 2015 Path
of
Mat
hs
and
Ori
gam
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th
e p
rim
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sch
ool
M.L
. Sp
reaf
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Dip
. S
cien
ze M
atem
atic
he,
Pol
itec
nic
o d
i T
orin
o
Some other remarks: blue rectangle have
same area
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Overlapping sheets:
the area difference depends on the CT only.
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The final model: a fractal (Model : Kiumars Sharif ; Diagram : Ali Bahmani © July. 2013 - Tehran, Iran)
Using this model we can touch the power’s properties
3 = 31
3x 31 = 32
3x 32 = 33
1 ? 1= 30
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Conclusions
Children
•obtain a deeper mathematical understanding •improve their communication skills •become increasingly precise in folding •enjoy this hands-on activity
and hopefully will have a better long-term
retention.
Didactics and Research of Folding – Freiburg 2015 Pat
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Thank you for your attention!
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Didactics and Research of Folding – Freiburg 2015
