Promoting Mathematical Thinking
  • Promoting Mathematical Thinking ... the home of John Mason & Anne Watson
    • IMP
    • PMThetaAtHome
    • Anne Watson >
      • Anne's PodCasts
      • Publications
      • Concerns
      • ATM triangles
      • Dose of Don
      • ToulouseGrids
    • AW & JM Joint Presentations
    • John Mason >
      • JHM Bio
      • Piranesi Inspired Pictures
      • JHM Presentations
      • Studies
      • Other Projects
      • JHM Publications
      • JHM-Draughty-Drafts
      • Applets >
        • Reasoning Without Arithmetic
        • Number Formats
        • Reasoning about Numbers
        • Reasoning about Graphs of Functions
        • Fractions & Decimals
        • More or Less Grids
        • Calgary Grids
        • Related Polygons
        • Polynomial Studies
        • Polygonal Projections
        • Sundaram Grids
        • Geometrical Applets
        • Structured Variation Grids
        • Animated Situations
        • Outer Inner Area Ratios
        • Clocks
        • Balancing Tasks
        • Linear Algebra
Picture

Number Formats

A collection of applets which focus attention on different ways in which Numbers can be formatted.
​The idea is that early familiarity with multiple formats is likely to enrich learners' sense of the structure of number


Relevant Apps on other pages:

​Drakensberg Grids

Numbers as "lots of" "lots of"...


Inspired by Stephen Yorgey's 'necklace' numbers, and Stephen von Worley's animated sequence of these, I constructed an applet that gives individual control over the number being formatted and the order in which the primes are used.

​They may be useful for immersing learners in the multiplicative structure of numbers

Notes are included with the applets.

Applet: "Lots of" Numbers
       ( File has to be unzipped!)
Numbers can be shown in sequence, or generated at random;

​Different orders of prime factors gives different diagrams.

Also included are applets for counting in different bases, for displaying the GCD (HCF) and LCM of a pair of numbers, and for comparing the collection of all factor trees (as a single factor lattice) and paths within that lattice with the different ways of displaying "Lots of" for a given number.

Picture
"Lots of" Diagrams for 12 (above) and using 4's when possible (below)
Picture

"Lots of" Displays for Different Bases

12 presented in base 3
Picture
​The use of different bases arises quite naturally in the following form:

You have cards with symbols from some otherwise unknown language, showing 0, 1, 2 and 3.

You have a collection of counters, and you need to inform someone who reads and understands only this other language, how many counters you have.

​The task is inspired by Zoltan Dienes, and refined by many people, including Tony Wing (UK) and Kieren Egan (Canada).

​Applet: Necklace Bases see applets in "Lots of"

Numbers as arrays of counters

Numbers as Rectangular Arrays
Numbers as Differences of square numbers, differences of triangular numbers, differences of similar rectangular numbers
Numbers as sums of at most 3 triangular numbers
Numbers as sums off at most 4 square numbers

APPLET: Number Formats
Picture

Numbers as Polygons

Some numbers arise as the number of counters in a layer of regular polygons, either arising from a vertex or centred around a central point.

Applet: Polygonal Numbers
Picture
Cornered and Central Polygonal numbers with 6 layers (above)
​ and with rays added (below)
Picture

Divisor Lattices and Thread Counts

Picture
The lattice of divisors of a number is an amalgam of all the factor trees:
two divisors of N re joined by an edge if and only if the larger divided by the smaller is a prime. 
The divisors fall naturally into layers, and the whole has an important structure.
​
The lcm and gcd (hcf) of two divisors can be found in the lattice. Indeed lattices are an abstraction of the property the each pair of divisors have an lcm and a gcd (hcf).

​

An important principle of counting arises in counting the number of threads from top to bottom in uniform diagrams like these corresponding to 12.

There are connections with necklace diagrams.
Picture
the Applet: Divisor Lattices provides the divisor lattice for. number, the various ways in which a thread count gives that number of threads, and a bipartite graph with the same number of edges.
Proudly powered by Weebly