Category Archives: Algebra II

Spaghetti Trigonometry

This lesson was adapted from the NCTM website.

The aim: For students to understand the origin and characteristics of the graphs of sin(x) and cos(x).

Student comments during the lesson:

  • This was the best lesson of the year!
  • Where did time go?
  • Can we have more lessons like this?

Image

Students will need:

  • Blank unit circle and trig. graph 
  • Glue stick
  • Protractor
  • A small pile of spaghetti noodles.

The student steps are simple:

  • Label the unit circle axis (-1 and 1’s) and the trig graph x-axis (0degs to 360degs). Do not yet label the y-axis.
  • Use a piece of spaghetti to mark a the unit distance from the origin at 15degs to the x-axis. Stick it down and label the angle.
  • Take another piece of spaghetti and measure the y-coordinate (sin(x)) of the point on the circle. Transfer this piece to lie on the trig graph vertically above the 15deg mark.
  • Repeat at 15 degree intervals all the way around the shape.
  • When you have finished, draw a line going over the top of all your spaghetti sticks to show the graph.
  • Finished all the way to 360degs? Try the same again but this time measure the x-coordinate (cos(x)) of each point on the unit circle (you will need a blank trig axis).

What I learned from teaching this lesson:

  • For general buy-in, it is really good for students to make predictions about how they think the graph will continue past 15 degrees, at the start of the lesson.
  • I certainly needed to demonstrate the first couple of noodles for students to get the idea, but they loved it.
  • I need to make more of the last 15 minutes of the lesson to really cement the learning, but this has given me license to talk about the graphs of sin(x) and cos(x) for some lessons to come.

It was very satisfying to show the animations found here in order to discuss the shapes of the curves.

Why I love Tarsia

Apparently it takes 10,000 hours to become an expert at anything you put your mind to, although not everyone agrees. A great pianist, however talented, does not become a virtuoso overnight. What is clear is that to become proficient in mathematics, yes, we need rich tasks and activities to build connections but sometimes you just need to ‘practice the scales.’ Many lessons out there are full of worksheets and textbooks where students can practice the same idea over and over again to their hearts content. The trouble is, this can be so boring for the individual and really demotivating for learning.

I want my mathematics classroom to be a place where students can get the practice they need but not get bored. That is why I love Formulator Tarsia.

Endless questions are boring, but put those questions in the context of group-work (possibly including competition) and suddenly it becomes a project where the task is to build a shape and the students find they are learning by accident. This is the best type of learning.

A download of this crucial program can be found here.

Pre-made files can be found here

I recently made a Tarsia activity for finding common denominators of rational expressions; this can be found here (right click and ‘Save Link As’).