Reasoning about Numbers
See also applets under the heading Fractions & Decimals
Number Line MovementsInitially this is a virtual version of walking forwards and backwards on a number line, in order to emphasise arithmetic as the study of actions of numbers on numbers.
With the addition of T (for turn around) learners can begin to reason about the actions without recourse to doing it physically, and eventually, without enacting it virtually. Download applet (with notes) 
Multiplicative Reasoning: The Drakensberg Grids
Going with the grain (down a column) provides variation of numbers.
Going across the grain (along a row) provides variation of presentation.
A second app presents a single row so that scaling is much larger.
Notes on applet use are provided.
Drakensberg Grid
Going with the grain (down a column) provides variation of numbers.
Going across the grain (along a row) provides variation of presentation.
A second app presents a single row so that scaling is much larger.
Notes on applet use are provided.
Drakensberg Grid
Varied Multiplications
What is the same and what is different about these five calculations? Why? How might they be extended to the rest of the grid? The applet permits the numbers to be changed, and also reveals reasoning behind the phenomenon 

Sharing OutIf you have ever picked berries, or shared things out as a child, you may recognise "3 for me and 2 for the pot" or "3 for me and 2 for you"!
What sorts of mathematical questions can you ask about this sort of situation? In how many different ways can you express the relationships between the number of blue, green and total after the process has been going on for an unspecified length of time? The idea is to introduce or to work on relationships between the language of fractions and decimals, both seen as actions on a set of objects. The applet permits choices of from 2 to 5 colours, in integer ratios. 
Set RatiosGiven a number of counters, in how many different ways can you place them in the three regions formed from two overlapping Circles (sets A and B) so that there are the same number of counters in both A and B?
What if the ratio is to be 3:2 for A:B? and more generally? What if there are three sets?

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CoveredUp SumsA selection of cells from a grid have been selected (think of the other cells as having been coveredup). Their sum is shown.
Make another selection, one cell from each row and columns. What is their sum? What is going on? What happens if you coverup all but two cells in each row and column? The applet supports constructing your own grid with a specified sum. What matters is the reasoning as to why, and why the construction works!

Reasoning About the Number Line
Using to the left of, to the right of, greater than and less, and also closer to ... than to ... to work out a number on the number line. One deals with whole numbers; one deals with fractions, and one deals with decimals. One uses clues of the form "the number is ... much more than an integer multiple of ..." where the divisors can be chosen to have denominators from 1 to 10. Still an early draft! Applets 
Screen shots from the various applets
Whole Number Searching
Decimal Searching
Fraction Searching
Remainder Reasoning
Number Line Reasoning
