Fig1.1
Fig1.2
Pineapples have 8 shallow spirals and 13 steeper spirals. The pine cone, above, has 8 shallow and 13 steeper spirals. Many pine cones are 5 by 8.
STEP 1: COUNTING THE SPIRALS AND MEASURING THE ANGLES
In Step 1, the students gain some experience with
(a) counting the spirals in various
plants (eg., pineapples, pine cones, or seed-heads of sunflowers or other flowers
such as asters or bee balm).
(b) examining the generative spiral
in cases where it can be seen; eg., branches or buds around the stem, or twigs
around a branch, or the spiral of petals in an artichoke.
COUNTING THE SPIRALS
To get this activity started, the teacher could use a pineapple or pine cone, or some other plant, to show how to count the spirals. To count the spirals in the case of a pineapple, establish a starting point by selecting a suitable scale at the top of the pineapple and marking it -- maybe by coloring it or sticking a pin in it. The spiral ending in that scale will be spiral number one. Now it is easy to count the spirals.
Fig1.1
Fig1.2
Pineapples have 8 shallow spirals and 13 steeper spirals. The pine cone,
above, has 8 shallow and 13 steeper spirals. Many pine cones are 5 by 8.
Fig.
1.3Fig. 1.4
The two sunflowers, above, are about 4.5 inches in diameter. These are
dwarf sunflowers with large headsThe first one has 55 spirals going counterclockwise
inward, and 89 clockwise. The spirals are a bit hard to count from the
photographs because the head of the flower is somewhat dome-shaped, and because
there is an overexposed area in the first photograph. The second one was
growing next to the first, and had the same general appearance, and is 55 by
89 but in the reverse direction. The second photograph was taken in flatter
light, making the spirals easier to count.
Fig.
1.5
Here is a sunflower from the same group, photographed two or three weeks later.
Some of the florets have fallen off, revealing the seeds underneath.
Fig.
1.6
This sunflower was growing in the same group. Whether it is the same kind,
I do not know. It has 34 shallow spirals counterclockwise inward, and
55 steeper spirals clockwise.
Fig. 1.7 
Fig. 1.8
Above is the seedhead of a giant sunflower, about 9 inches in diameter, with
89 spirals clockwise inward and 144 counterclockwise. Someone got it on
a trip to North Dakota and brought it to class. On the right is a close-up of
the same sunflower.
Fig. 1.8
Fig. 1.9
Here are some black-eyed susans going to seed. The seed-head
has 13 shallow spirals and 21 steeper spirals.