Applets & Animations

revised June 2013

(New Pebble Arithmetic Animations)


1 Reasoning w/o Arithmetic 2 Function Studies 3 PolyDials4 Animated Triangles5 Linear Algebra6. Calculus7. Infinite Sums8. Pebble Arithmetic9. Ratios & Scaling10. Three Pts Determine ...11. Carpet Theorem ...12. Algebraic Generalisations13. Exchange

 Ways of Working

My 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 Arithmetic ...

A collection of situations in which learners young and not so young can reason mathematically without requiring arithmetic fluency.

Secret Places (homage to Tom O'Brien): Secret Places; notes; online applet
    there is a 2D version inside the zipped folder using surfaces such as a cylinder, mobius band, torus, and Klein bottle

Magic Square Reasoning: Reasoning based on the properties of magic squares but wihtou ever referrin to a particular magic square. PPT

Revealing 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 this. (RevealingTheShapes) If you 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 @ 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)

More-or-Less 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.
  More-or-Less perimeters and areas

 2. Function Studies

ZIGZAGs: initial stimulus to exploring absolute value function through first generating and then characterising zigzag functions of the form |||x-1|-2|-3|. Animation1; Animation2

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.

Animation; Cabri page1; Cabri page2

Lagrange & Tangents

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.
Iter1; Iter 2; Iter3

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)

Explaining 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 download; Ceiling - floor web ;


Functions with interesting properties concerning differentiability at x = 0. Floor-Function Download; Floor-Fn web;

3. PolyDials

Polydials 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.

DownLoad PolyDials 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 Triangles

iInscribed Squares

This applet is about inscribing squares in a triangle.  It is in early draft form and will have more extensions. Web version; Download

5. Linear Algebra

Designed to show how a linear mapping works, it leads to Eigen-directions and beyond. See Notes in folder   DownLoad Linear Mapping
(zipped folder: open html file; keep all three files together)

6. Calculus

7. Infinite Sums

Two animations showing diagrammatically the sum of a GP with first term a/7 and ratio a/7 for a = 1, 2, 3; Sevenths. A 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 infinite sum.

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 of films 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

9. 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; Compound Scaling; 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!

10. Three Points Determine ...

Through exercising 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 applet

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; notes; online applet
  Areal Relation: download;
notes; online applet 1; online applet 2

  Archimedes Salinon: download; notes; online applet 1; online applet 2

12. Algebraic Generalisations

Patterns from 2: Variations on the observation that 2 + 2 = 2 x 2 and its generalisations. Patterns from 2

13. Exchange

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