Download Exploring example spaces 2006

Exploring example
spaces:
what are they like and
how do we move around
them?
Anne Watson
Jasper, October 2006
2,4,6,8 …
5,7,9,11 …
9,11,13,15 …
► Make
up a similar sequence of your own for
which your neighbour will find the sum of
the first five terms.
2, 4, 6, 8 …
2, 5, 8, 11 …
2, 23, 44, 65 …
► Make
up a similar sequence of your own for
which your neighbour will find the sum of
the first five terms.
2,4,6,8 …
3,6,9,12 …
4,8,12,16 …
► Make
up a similar sequence of your own for
which your neighbour will find the sum of
the first five terms.
Principle 1
► All




learners have a natural propensity to
see patterns,
to seek structure,
classify,
generalise ….
2,4,6,8 …
5,7,9,11 …
9,11,13,15 …
2, 4, 6, 8 …
2, 5, 8, 11 …
2, 23, 44, 65 …
2,4,6,8 …
3,6,9,12 …
4,8,12,16 …
Principle 2
► Example
spaces can be characterised by
their dimensions of variation and ranges of
of change
The largest …
► Sketch
a quadrilateral whose sides are all
equal in length. Area?
► Sketch a quadrilateral for which two pairs of
sides are equal in length, and which has the
largest possible area.
► Sketch a quadrilateral for which three lines
are equal in length, and which has the
largest possible area.
► … same for no lines equal
Principle 3
► Constraints
make the problem more
interesting/ harder/ more conceptual
► Write
down a pair of numbers which have a
difference of 2
► ….. and another pair
► ….. and another pair
Principle 4
► Example
spaces are individual, and learners
can be prompted to extend their example
spaces
► Write
down a pair of numbers which have a
difference of 9
► ….. and another pair
► ….. and another pair
► On
a nine-pin geoboard, create a triangle which
has a height of two units.
 and another
 and another
► Using
dynamic geometry software, find the class
of triangles which have a height of two.
► Construct
a triangle which has a height of two
and a height of one.
 and another
 and another
Principle 5
Example spaces are dependent on context
and tools
Example of what?
Principle 6
► Examples








have to be examples of something:
classes of objects
concepts
techniques
problems and questions
appropriate objects which satisfy certain conditions
ways of answering questions
ways to construct proofs
…. so on
Sorting examples
► Think
of a number
► Add 3 to it and also subtract 3 from it;
also multiply it by 3 and divide it by 3
► Now put your four answers in increasing order, and
label then with their operations
► If you change the 3 to something else, is the order
always the same for your starting number?
► If you change your starting number, but preserve 3,
what different orders can you achieve?
► What if you change both the starting number and
the 3?
Principle 7
► You
can explore and extend your example
spaces by:




sorting
comparing
combining
… what else?
Can you see any fractions?
Can you see 1 ½ of something?
Principle 8
► The
process of creating examples is
dependent on the way it is prompted
Examples of methods
► Think
of as many ways as you can to
enlarge a rectangle by a scale factor of 2
►
Sequences:
 what does “like this” mean?
 we all look for patterns
►
Quadrilateral
 start from what we know and make it harder by adding
constraints
►
Difference of 2
 ..and another – push beyond the obvious
►
Triangle with height 2
 fix properties to encourage play with concepts
►
Grid - of what?
 similarity as a tool, or as a muddle?
►
Use 3 to +, -, ×, ÷
 Using learners’ own example spaces to sort, compare,
relate …
►
Seeing fractions
 open/closed questions
►
Enlarging rectangles
 shifting to more powerful methods
Principle 1
► All




learners have a natural propensity to
see patterns,
to seek structure,
classify,
generalise ….
Principle 2
► Example
spaces can be characterised by
their dimensions of variation and ranges of
of change
Principle 3
► Constraints
make the problem more
interesting/ harder/ more conceptual
Principle 4
► Example
spaces are individual, and learners
can be prompted to extend their example
spaces
Principle 5
Example spaces are dependent on context
and tools
Principle 6
► Examples








have to be examples of something:
classes of objects
concepts
techniques
problems and questions
appropriate objects which satisfy certain conditions
ways of answering questions
ways to construct proofs
…. so on
Principle 7
► You
can explore and extend your example
spaces by:




sorting
comparing
combining
… what else?
Principle 8
► The
process of creating examples is
dependent on the way it is prompted
If I had to describe my conclusion [as to a
method of studying] in one word, I’d say
examples. They are to me of paramount
importance. Every time I learn a new
concept I look for examples … (Halmos 1985).
On arriving at any new rule or process, the
student should work a number of examples
sufficient to prove to himself that he
understands and can apply the rule or
process in question…. He may choose an
example for himself, and his previous
knowledge will suggest some method of
proving whether his result is true or not. (De
Morgan 1831).