Spaghetti Sine Graph Activity

Hands-On Activity

Spaghetti Sine Graph

Building \(y = \sin(\theta)\) from the Unit Circle

Unit Circle Sine Values Radian Measure Graphing Periodic Behavior

Setup

Required Materials

📄

Paper

2 sheets, taped long-ways

🍝

Spaghetti

Uncooked, stiff

🧵

String

~12 inches

🖊️

Marker

Any color

📏

Ruler

For straight lines

🥣

Bowl

For tracing circle

Step 1

Prepare Your Unit Circle

Draw the Circle

Trace a bowl and draw your axes through the center.
Label the origin \((0,0)\) and rightmost point \((1,0)\).

Estimate Angles Using Clock Positions

\(0 \approx 3\) o'clock
\(\frac{\pi}{6} \approx 2\) o'clock
\(\frac{\pi}{3} \approx 1\) o'clock
\(\frac{\pi}{2} \approx 12\) o'clock

Clock Reference

\(0\) \(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{3}\) \(\frac{\pi}{2}\) \(\frac{2\pi}{3}\) \(\frac{3\pi}{4}\) \(\frac{5\pi}{6}\) \(\pi\) \(\frac{7\pi}{6}\) \(\frac{5\pi}{4}\) \(\frac{4\pi}{3}\) \(\frac{3\pi}{2}\) \(\frac{5\pi}{3}\) \(\frac{7\pi}{4}\) \(\frac{11\pi}{6}\) \(2\pi\)

Continue Around

Mark all angles from \(0\) to \(2\pi\) using the same clock-position strategy. Label each tick with its radian value.

Step 2

The String & Graph Paper

The String

Wrap string around the circle. Mark it at each angle (\(0\) to \(2\pi\)). This becomes your \(\theta\)-axis.

The Paper

Tape two sheets long-ways.
Draw a horizontal axis labeled \(\theta\) and a vertical axis for \(y = \sin(\theta)\).
Transfer string marks to the horizontal axis to get evenly-spaced angle positions.

Result: Your Theta Axis

unrolls to

\(0\) \(\frac{\pi}{2}\) \(\pi\) \(\frac{3\pi}{2}\) \(2\pi\)

Step 3

Measuring with Spaghetti

Measure the Height

Hold spaghetti vertically from the \(x\)-axis to the point on the circle.
Break it at that exact height -- this is the \(\sin(\theta)\) value.

Transfer to Graph

Transfer the piece to your graph paper at the matching \(\theta\) mark.
Repeat for every labeled angle from \(0\) to \(2\pi\).

Your Graph Should Look Like This

\(0\) \(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{3}\) \(\frac{\pi}{2}\) \(\frac{2\pi}{3}\) \(\frac{3\pi}{4}\) \(\frac{5\pi}{6}\) \(\pi\) \(\frac{7\pi}{6}\) \(\frac{5\pi}{4}\) \(\frac{4\pi}{3}\) \(\frac{3\pi}{2}\) \(\frac{5\pi}{3}\) \(\frac{7\pi}{4}\) \(\frac{11\pi}{6}\) \(2\pi\)

Step 4

Building the Wave

Direction Matters

Positive sine values: pasta goes UP from the axis.
Negative sine values: pasta goes DOWN below the axis.

Connect the Tips

Connect the tips of the pasta with a smooth curve. You've turned circle heights into a sine wave!

The Result

Reflect

Final Reflection

Moving around a circle creates repeating vertical motion -- a wave.

Question 1

Where is \(\sin(\theta)\) at its maximum? What angle? What value?

Question 2

Where does \(\sin(\theta)\) equal zero? How many times per cycle?

Question 3

Where is \(\sin(\theta)\) at its minimum? What angle? What value?

Key Idea

\[\text{Circle} \;\longrightarrow\; \text{Wave} \;\longrightarrow\; y = \sin(\theta)\]