personal preferred way of working with animations and applets is in plenary. With animations, I like to
show the annimation once and then invite personal and paired
reconstruction of the incidents. This usually leads to differences or
to gaps which inform a second viewing. Once there is an agreed narrative, attention turns to accounting-for
what happens mathematically.
With other applets, I try to provoke mathematucal thinking and then show how the applet can be used to explore more deeply.
1. Reasoning without
A collection of situations
in which learners young and not so young can reason mathematically
without requiring arithmetic fluency.
Places (homage to Tom O'Brien): Secret
there is a 2D version inside the zipped folder using surfaces such as a
cylinder, mobius band, torus, and Klein bottle
Reasoning: Reasoning based on the properties of magic squares but wihtou ever referrin to a particular magic square. PPT
Shapes: The applet provides Opportunities for reasoning, following consequences of
possibilities and so bringing to awareness that choices have
consequences. (This applies in the social realm as well as in
mathematics.) Experience suggests that quite young children can manage
generate an interesting example and you want it included in the applet,
send me a picture and I will build it in (j.h.mason @ open.ac.uk). The
current examples are visible here
The applet is designed to be downloadable
(unzip the folder and then open the html file in a web browser)
is a task structure originated by Dina Tirosh and Pessia Tsamir. The
idea is to present a grid and ask students to construct vraiations on
the object in the central cell which vary one or both of two possible
attributes. A related structure exploited by Colin Foster is to provide
a grid with attributes as row and column entries and then invite
students to construct one or more objects in each cell.
initial stimulus to
exploring absolute value function through first generating and then
characterising zigzag functions of the form |||x-1|-2|-3|. Animation1;
Cobwebs provides a stimulus to
learn to read graphs by tracking
coordinates to discover a locus and that composition is not
commutative. Download Zipped Folder
The applet makes it easier to
check (by eye) that various alignments seem to be inevitable. There is
the challenging task of relating two polynomials so that both their
composites display the appropriate degree!
See Notes for questions, prompts and explanation of buttons.
CHORD STUDIES: stimuli to
think about the relationships between the
chord-slope function where one end of the chord is kept fixed, and the
slope-chord function where the interval width of the chord is kept
fixed, and then allowed to approach 0.
CUBIC CHORDS: opportunity
to explore the surprising fact that the family of chords on a cubic
which all have the same x-coordinate for their midpoint, when extended,
all intersect on the cubic again at the same point. Cubic
Chords web version; Download
(unzip then open html file)
COBWEBS: Application of
compound maps to iterations,
including a cyclic iteration.
PHASE DIAGRAMS &
Phased by Distance: given two points in the plane, measure the distance
from P to each and then use
these as coordinates to graph the position of P.
Phased by Distance
CEILING - FLOOR (Nov 5 2010)
why the graphof ceiling(x-a) - floor(x-b) looks the way it does, and
how to force the segments to be equal in length. Ceiling - floor
- floor web ;
INTERESTING FLOOR FUNCTIONS
Functions with interesting
properties concerning differentiability at x = 0. Floor-Function Download; Floor-Fn web;
is an applet under
construction with Darren Peirce
(programming). It will be an example of how icons can be used to
express as-yet-unknowns to create an algebraic analysis of a complex
situation. The context is a vast generalisation of classic
'flipping cups' and Chinese Jigsaws as an exploration of group theory
connected with modular arithmetic.
folders (need unzipping)
The applet is rough as yet:
files have to be opened directly or you can
construct your own. For further information contact me.
4. Animated Triangles
This animation was made in
geogebra for me by James Robinson
for use in sessions reviewing area of triangles. DownLoad Animated
This applet is about inscribing squares in a triangle. It is in
early draft form and will have more extensions. Web
Designed to show how a
mapping works, it leads to
Eigen-directions and beyond. See Notes in folder DownLoad
(zipped folder: open html file; keep all three files together)
7. Infinite Sums
animations showing diagrammatically the sum of a GP with first term a/7
and ratio a/7 for a = 1, 2, 3; Sevenths.
GP with first term 1/8 and ratio 1/8 for a = 1 , 2, 3, 4.
These can be seen as generic for any fraction; Eighths.
Notes on an
8. Pebble Arithmetic
These animations were inspired by a
comment in Bloor's book (Knowledge
& Social Imagery 1976) in which he makes use of an
idea of J. S. Mill called Pebble Arithmetic, inspired by an idea of Dienes (The Power of
Mathematics 1964). Film1 suggests a generalisation which, when
expressed, gives an algebraic identity equivalent to Dienes' idea.
There is a set
with notes on ways of working with them, each one of which invites a
generalisation, some of them as a sequence invite a further
generalisation, and all of them together invite yet a further
generalisation, all to do with factoring.
Film2 and Film3
invite the question, if somone
claims to have a sequence derived in
this way, can you reconstruct the number of pebbles and the starting
configuration, or else know that that sequence is impossible?
NB: There is now (June 2013) a full set of animations in a pptx format (Pebble PPT), a draft set of notes with embedded animations (Notes) and a set of applets (Pebble Applets)
that may or may not work depending on a browser that will actually run
Java (open the html file in a java-enabled browser). Currently,
java applets only work from a non-local server, so you may have
to use them from here: JSMill Original; Pebble Arithmetic Sequences; Triangular Pausing Pebble Arithmetic
Ratios & Scaling
These applets are intended
to display the way scaling of a figure
works, and how combining two scalings produces a third with the scale
factor being the product of the scale factors. Multiplication is
quintessentially scaling, and 'repeated addition' is a special
instance when the scale factors are integers. Elastic Multiplication;
Polygon Scaling; Duck
Weed (exponential thinking)
I use these after showing people
how an elastic can be stretched. In
particular, mark the midpoint and the one-third point along an elastic.
Now stretch (hold one end fixed and pull the other) until the
one-third-point is where the midpoint was originally. By what
factor has the elastic been stretched? Generalise!
Three Points Determine ...
mental imagery and drawing upon the mathematical theme of freedom and
constraint, it is not difficult to build up to the theorem that three
points determine a circle (measure the freedom under constraints such
as 'circle passes through 1 pt', then '2 pts' then '3 pts' of the
centre of the circle). The applet then provides supportive imagery. The
applet can also be used to explore the questions of 'how many points
determine a square' (and what constraints might be needed) and 'how
many points determine a rectangle. 3 Points Determine
11. Carpet Theorem Applied
The carpet Theorem says
that if you have two overlapping carpets, and you move them, the change
in the area of overlap is the change in the area not covered by the
carpets. A simpler version is that if you have two carpets exactly
covering the floor of a room without overlap, and if they are moved so
as to overlap, then the area of overlap is the area of bare floor.
This observation can be applied in many situations. For example,
Circles in Circles: download;
Areal Relation: download; notes; online
applet 1; online
Archimedes Salinon: download; notes; online
applet 1; online
12. Algebraic GeneralisationsPatterns from 2: Variations on the observation that 2 + 2 = 2 x 2 and its generalisations. Patterns from 2
Set Ratios: Placing a set number of objects into two or three overlapping sets so that the numbers in the sets are in a specified ratio. Set Ratios Applet