1. Mathematical Environments
Jennifer Piggott

2. Outline
The session will involve investigating some fruitful mathematical
environments.
We will work on problems and environments available on the
NRICH website and will examine their potential to meet the needs
of a range of curriculum contexts.
We will look in detail at two or three examples based on:
Isosceles triangles
Circular Geoboards
Cuisenaire rods

3. Making Rectangles – Making Squares
You have 20 equilateral triangles all of the same size as well as the same number of 30°, 30°, 120° isosceles triangles with the shorter sides the same length as the equilateral triangles.
Using these triangles how many differently shaped rectangles can you build? Can you make a square?
What other questions does this environment invite you to ask?
Published:April 2001.

4. More questions How many triangular n-animals can you make?
(Triangular animals are polyominoes made with equilateral triangles).
Create a shape using the two types of triangles and calculate the fraction of the shape made with each type of triangle.

5. Circular Geoboards Nine-Pin Triangles (July 2005)
How many different triangles can you make on a circular pegboard that has nine pegs?
Triangles all around (July 2005)
How many different triangles can you draw on a circular pegboard which has four equally spaced pegs? What are the angles of each triangle? If you have a six-peg circular pegboard, how many different triangles are possible now? What are their angles? How many different triangles could you draw on an eight-peg board? Can you find the angles of each?

6. Triangle Pin Down (July 2005)
A right-angled triangle has been drawn on the four-pin board. Can you draw the same type of triangle on a three-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? What kind of triangle is drawn on the six-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? Can you name the type of triangle drawn on the nine-pin board? On what size board could you draw the same type of triangle? Do you notice anything about the number of pins for which this is possible?

7. Other problems in July 2005 Triangle Pin down
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Triangles in Circles
How many different triangles can you make which consist of the centre point and two of the points on the edge? Can you work out each of their angles?

8. Board Block Challenge (July 2005)
Firstly, choose the number of pegs on your board. Decide what shapes you will be allowed to make. You could allow:
triangles and quadrilaterals
triangles, quadrilaterals and pentagons …
Take it in turns to add a band to the board to make any of the shapes you are allowing. A band can share a peg with other bands, but the shapes must not overlap (except along the edges and pegs). A player loses when they cannot make a shape on their turn. For your choice of shapes, how does the winning strategy change as you increase the number of pegs on the board?
Board Block
Firstly, choose the number of pegs on your board. Take it in turns to add a band to the board. Bands must fit round three pegs, in other words, each must make a triangle. A band can share a peg with other bands, but the triangles must not overlap (except along the edges and pegs). A player loses when they cannot make a triangle on their turn.

9. Other Problems from July
Subtended Angles
Right Angles
Pegboard Quads
Sine and Cosine for Connected Angles
Subtended angles
Stage:3 Challenge Level:
What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
Right angles
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
Pegboard Quads
Stage:4 Challenge Level:
Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?
Sine and Cosine for Connected Angles
The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

10. Squares Square it (Oct 2004) Tilted Squares (Sept 2004)
Square coordinates (Feb 2005)
A tilted Square (Jan 2003)
Need combining into a single environment

11. Cuisenaire (Oct 2005) Train game
This is a game for two players. You need one Cuisenaire rod of each length between 1 (white) and 10 (orange), or you can just write down the numbers on a piece of paper. Decide who is going to go first, and choose a distance between 11 and 55.

12. Other resources (Oct 2005) Colour building Different by one
Rod fractions
General environment

13. November - Probability
Spinners
Epidemic modelling
Simple probability environments
Articles

14. For slides and more
Search for:
Rochdale 2005