Lesson 6

Sketching the Sundial

Lesson created by: Chris Copeland

Grade: Pre-Calculus

Length: Approximately 3 days

Goal: Students will use Geometer’s Sketchpad to understand and explore several geometric properties behind the sundial, as well as to sketch an accurate design for their sundial. Assume prior knowledge of how to work sketchpad.

Objectives: Students will be able to use Geometer’s Sketchpad to construct a blueprint for their sundial. They will learn how to use sketchpad to explore concepts in geometry and pre-calculus.

TEKS:
111.35.c.3: The student uses functions and their properties to model and solve real-life problems. The student is expected to:
(A)  use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;
(D) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas.

Engage: Have you ever looked at a theorem, construction, or property shown in class and said to yourself, “I don’t really see the big picture from this”? Not being able to visually see what is being talked about can sometimes be a real road block for understanding the concept, so today we’re going to analyze in sketchpad some of the properties that we’ll be using for building our sundial.

Explore: First let’s look at this whole sine, cosine, and tangent thing. Construct any triangle in sketchpad that you want, and label the vertices A, B, and C any way you want. Click measure -> angle for one angle, and then calculate the cosine for it.

-Question: But wait! I thought we could only find sine & cosine for right triangles?

-While this is true, is there any way to turn this triangle into a right triangle while keeping the chosen angle the same measure? Do so.

From this smaller right triangle, measure the distance of the adjacent side to the angle and the hypotenuse (adjacent / hypotenuse = cosine). Does the value for the ratio of the side measures equal the answer you got for the cosine of the angle? Well that’s exactly how sine and cosine are calculated without regard to whether the triangle is right or not.

Recall the law of sine’s. On your own, try to show that the law of sine’s works.

Do the same to test properties of arcs (circumference / arc length = 360 / arc measure), and show that the point of intersection of two perpendicular bisectors of any chords on a circle meet at the center.

Explain: For each exercise, write a brief explanation in your own words as to how you think the property works after having examined it in sketchpad.

Elaborate: What can you determine about inverse trig functions from using sketchpad?

Evaluate:

Now let’s consider the base of our sundial. The most common approach is to make it circular, so let’s try to accurately design it on sketchpad.

The first step will be to construct a circle and know its radius. Since sketchpad automatically lists its measurements in centimeters, make a key to convert what the length of your actual sundial base will be compared to the sketch. Include the angle measurements for the hour marks [tan(x) = tan(hour angle) * sin(latitude), where x = the angle from noon]. If you’re going to turn your sundial into a different geometric shape, remember to keep the circle in the sketch. Later, we’ll sketch the gnomon.