1. In what different ways can you join the points? What types of figures do you get?

What do you get if each point is joined to the next to it, i.e. one point away from it? What happens if each point is joined to the point two away from it? Three away from it?

And so on. How do your results depend on the number of points?

2. Apply the [n,p] notation to your findings for a particular number of points in Hint 1.

What do you notice about the results? Are the results different for each value of p? What different values of p result in identical figures? Which figures start and finish at the same point, and involve every other point in the

process? Which figures involve overlapping figures? Which figures look like the spokes of a wheel? For what values of p can figures be drawn?

3. What do you notice about the number of different figures for a particular number of points?

Can you predict the number of different figures?

4. What do you notice about the number of chords in figures for a given number of points?

Can you predict the number of chords for a particular [n,p] figure?

STARS

Each of the six equally spaced points on this circle has

been joined to a point that is two points away from

it in a clockwise direction. The result can be called

a [6,2] figure.

Equally spaced points on a circle can be joined by chords in various ways. Investigate.

STARS

A. Aspects/problems considered

B. Methods of attack used

• Draw a diagram

• Look for a Pattern

If the six (6) equally spaced points on the circle has been joined to a point that is p point/s away in a clockwise direction, the result can be called an [n,p] figure. (n=number of points, p=points away)

p=0 p=1 p=2 p=3

p=4 p=5 p=6 p=7

p=4 p=5 p=6 p=7

Circle (p=0) 0 6 12 18 24 30 36 42

Hexagon 1 5 7 11 13 17 19 23

C. Data obtained

D. Patterns/relationships occurred

(p=1)Star (p=2) 2 4 8 10 14 16 20 22

Wheel’s spoke (p=3)

3 9 15 21 27 33 39 45

Circle p+6, p+6+6, p+6+6+6…

Hexagon p+4, p+4+2, p+4+2+4, p+4+2+4+2…

Star p+2, p+2+4, p+2+4+2, p+2+4+2+4…

Wheel’s Spoke p+6, p+6+6, p+6+6+6…

Number of Points Away Number of Regions Number of Chords0 1 01 7 62 13 63 6 34 13 65 7 6

From the series of observation, the following conjectures were made:

From the patterns observed in predicting what figure will be drawn for a particular [n,p] figure, the following conjectures were made.

If the number of points away is a multiple of six (6), then the figure that will be drawn is a Circle.

If the number of points away is a multiple of three (3) and is not a multiple of six (6), then the figure that will be drawn is a Wheel’s Spoke.

E. Conjectures made

Conjecture A

Conjecture B

Conjecture C

If the number of points away is even and is not a multiple of six (6) and three (3), then the figure that will be drawn is a Star.

If the number of points away is odd and is not a multiple of three (3), then the figure that will be drawn is a Hexagon.

From the patterns observed in predicting the number of the region/s for a particular [n,p] figure, the following conjectures were made.

If the number of points away is a multiple of six (6), then the number of region of the figure is one (1).

If the number of points away is a multiple of three (3) and is not a multiple of six (6), then the number of region of the figure are six (6).

If the number of points away is even, and is not a multiple of six (6) and three (3), then the regions of the figure are thirteen (13).

If the number of points away is odd, and is not a multiple of three (3), then the regions of the figure are seven (7).

From the patterns observed in determining the number of chords for a particular [n,p] figure, the following conjectures were made.

If the number of points away is a multiple of six (6), then the number of chords of the figure is zero (0).

If the number of points away is a multiple of three (3) and is not a multiple of six (6), then the number of chords of the figure are three (3).

Conjecture D

Conjecture E

Conjecture F

Conjecture G

Conjecture H

Conjecture I

Conjecture J

Conjecture K

If the number of points away is even or odd and is not a multiple of six (6) and three (3), then the number of chords of the figure are six (6).

If the number of points away is a multiple of six (6), then the figure that will be drawn is a Circle.

To test this conjecture we should try the values of p (points away) which are multiples of six (6).

p=12 p=30 p=42

The conjecture has been supported but not yet justified.

If the number of points away is a multiple of three (3) and is not a multiple of six (6), then the figure that will be drawn is a Wheel’s Spoke.

To test this conjecture we should try the values of p (points away) which are multiples of three (3) and is not a multiple of six (6).

F. Testing the Conjectures made

Conjecture A

Conjecture B

p=15 p=27 p=45

The conjecture has been supported but not yet justified.

If the number of points away is even and is not a multiple of six (6) and three (3), then the figure that will be drawn is a Star.

To test this conjecture we should try the values of p (points away) which are even and are not multiples of six (6) and three (3).

p=10 p=20 p=22

The conjecture has been supported but not yet justified.

If the number of points away is odd and is not a multiple of three (3), then the figure that will be drawn is a Hexagon.

To test this conjecture we should try the values of p (points away) which are odd and are not multiples of three (3).

p=11 p=13 p=17

The conjecture has been supported but not yet justified

If the number of points away is a multiple of six (6), then the number of regions of the figure is one (1).

Conjecture C

Conjecture D

Conjecture E

To test this conjecture we should try the values of p (points away) which are multiples of six (6) and should have a figure that has one (1) region.

The conjecture has been supported but not yet justified.

If the number of points away is a multiple of three (3) and is not a multiple of six (6), then the number of regions of the figure are six (6).

To test this conjecture we should try the values of p (points away) which are multiples of three (3) and is not a multiple of six (6) and should have a figure that has six (6) regions.

The conjecture has been supported but not yet justified.

If the number of points away is even, and is not a multiple of six (6) and three (3), then the regions of the figure are thirteen (13).

Conjecture F

Conjecture G

To test this conjecture we should try the values of p (points away) which are even and odd are not multiples of six (6) and three (3) and should have a figure that has thirteen (13) regions.

The conjecture has been supported but not yet justified.

If the number of points away is odd, and is not a multiple of three (3), then the regions are seven (7).

To test this conjecture we should try the values of p (points away) which are odd and are not multiples of three (3) and should have a figure that has seven (7) regions.

The conjecture has been supported but not yet justified.

If the number of points away is a multiple of six (6), then the number of chords of the figure is zero (0).

Conjecture H

Conjecture I

To test this conjecture we should try the values of p (points away) which are multiples of six (6) and should have a figure that has zero (0) chord.

The conjecture has been supported but not yet justified.

If the number of points away is a multiple of three (3) and is not a multiple of six (6), then the number of chords of the figure are three (3).

To test this conjecture we should try the values of p (points away) which are even and odd are not multiples of six (6) and three (3) and should have a figure that has six (6) chords.

The conjecture has been supported but not yet justified.

If the number of points away is even or odd and is not a multiple of six (6) and three (3), then the number of chords of the figure are six (6).

Conjecture J

Conjecture K

To test this conjecture we should try the values of p (points away) which are multiples of three (3) and is not a multiple of six (6) and should have a figure that has three (3) chords.

The conjecture has been supported but not yet justified.

The multiples of six (6), as points away, are said to form a figure of circle since, if we are going to connect the six equally spaced points in a circle, say for instance 0 point away, 6 points away, 12 points away and the other multiples of six, the initial point will also be its terminal point. Thus, a figure of circle will be formed. This idea follows that the region/s and chord/s formed for the multiples of six (6) are 1 & 0 respectively.

The multiples of three (3) which are not multiples of 6, as points away, are said to form a figure of wheel’s spoke since, if we are going to connect the six equally spaced points in a circle, say for instance

G. Explanation for Conjectures made

Conjectures A, E & I

Conjectures B, F & K