Stars and factors

[Making Stars] [Counting Stars] [Some answers] [Division] [Workshops]

This activity provides a way to link geometry and an exploration of factors.

Making stars

To begin the class needs some stars.

For this activity a star includes the usual sort of stars:
and stars that are really pointy:
as well as polygons that don't really look like stars:

There are three approaches you might take to making stars.

One is to have the kids punch holes around the perimeter of a paper plate and string yarn through the holes to make stars.  You'll need paper plates with 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 holes.  You'll need half as many plates than the number of holes (i.e., you'll need 4 plates with 8 holes.)

Or you could give them worksheets with circles marked with dots.  Then they can draw lines between the dots to make stars.  You'll need circles with  3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 dots, and you'll need half as many circles as the number of dots (i.e., you'll need 4 circles with 8 dots.)  Actually you'll need more because the kids will mess some up.

Or you could use LOGO.  This procedure draws a star:

to star :sides :skip :size
cs
pu
fd :size/2
rt 90
rt (:skip*360/:sides)/2
pd
Repeat :sides [
fd :size
rt :skip*360/:sides
]
end
To use it the kids type STAR 7 3 100.  "7" can be any number, and is the numbers of points on the star.  "3" can be any number less than "7" but bigger than 0 except it cannot be a divisor of "7" (This is a bug in the programme that I am working on).  "100" is the size of the star, and values between 100 and 200 work best.

STAR 7 3 100 produces: 
 

Counting Stars

The number theory in this activity comes out of asking questions like:

Some answers

How many stars are there with n points?

 
Points
Number
3
1
4
2
5
2
6
3
7
3
8
4
9
4
10
5
The pattern is that the number of stars is half the number of points, rounded down.  Encourage the children to explain this.  They might make use of the number of holes or dots they skip in stringing their yarn or connecting their dots.  It is possible to make a star (a polygon) going to the next dot, skipping one, skipping two, etc. but when they reach the hole or dot directly across (for an even n) are almost directly across, all the remaining stars will repeat ones they already did.

Which stars are made of two or more pieces? and Which stars are made of lines?

Stars in which they are joining every h hole (or dot) if h is a divisor of n.
 
Points (n)
Skip (h)
Pieces
Picture
4
2
2
6
2
2
6
3
3
Three lines
8
2
2
8
4
4
Four lines
9
3
3
10
2
2
10
4
2
10
5
5
Five lines
12
2
2
12
3
3
12
4
4
12
6
6
Six lines

There are several patterns to notice here:  When h is half of n the star becomes h lines.  When h is 2 the star becomes two copies of a n/2 sided polygon.  When h is 3 the star becomes three copies of a n/3 sided polygon.  When h is 4 the star becomes four copies of a n/4 sided polygon. As all of these involve factors of n when n is a prime number the stars are all made of one piece.
 

More Stars

Here is an image bank of stars
 
Points
Skip = 1
2
3
3
   
4
 
5
 
6
 
7
8
9
(Skip = 4)


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Copyright David A Reid