Measuring latitude with the shadow of a pole is most rewarding, if one appreciates that this is precisely how the Greek scholar Eratosthenes of Cyrene, chief of the Library of Alexandria, was able to determine not only that the Earth was round, but to calculate its circumference in the third century B.C.E. Indeed, it was Eratosthenes who established latitude (as well as longitude).
At the library, Eratosthenes read a document describing how in Swenet, far to the south, a stick in the ground cast no shadow at noon on the longest day of the year. By measuring the length of a shadow cast by a stick of the same length in Alexandria, Eratosthenes determined that the two cities were separated by an arc of 7 degrees, or about 1/50 the curvature of a circle. Multiplying the distance between the two cities by 50, he calculated the Earth's circumference, within 1 percent of what we know it to be!
Find out the exact direction of north and south, and imagine a line running between the North and South poles all the way around the planet. This is a longitudinal line. When the sun crosses the longitudinal line, it is at its highest point in the sky for the day.
Spend several weeks observing the sun's movement to find the time when it is at its highest point. Ideally, this is at noon. Because time zones are very big, however, there are few locations where this occurs EXACTLY at noon. Being on or off daylight savings time ads another complication. Consequently, the time of day when the sun is at its highest could occur within a couple of hours before or after noon.
When the sun is at its highest point in the sky, measure the length of the shadow of a vertical pole, measure the height of the pole, and plug the results into the following formula: S = H tan( L-((tilt) * sin(360degrees t /365) ), where S is the shadow length, H is the height of the pole, tilt is the inclination of Earth's axis with respect to its orbit around the sun (23.5 degrees), where t is the number of days from the vernal equinox and L is the unknown latitude.