Category Archives: Trigonometry

Escape the Worksheet: Sidewalk Chalk

Mathematics students need to practice mathematical skills. That much is certain. What I am trying to do in my classroom is to get away from the here-are-20-questions-go activity that puts shivers down the spine of many students. So how do I still help them to practice but in a less monotonous way?

A simple idea, now that the weather has improved, is for the students to do the same problems but using sidewalk chalk to decorate the school’s pathways with beautiful math. There is something in this that appeals to the 3-year-old in all of us.

The lesson objective was clear: To prepare for the quiz on 1) Solving triangles and 2) Using the unit circle to find trig ratios. We wandered outside and went for it.


How it went:

  • Most students enjoyed the lesson with a handful opting to carry on with a paper worksheet. At the end, students fed back that they enjoyed working outside.
  • I loved the instant assessment. It took very little time to see what was going on in the students minds. It also kept the accountability high. It is difficult to a student to fake writing their work on a sidewalk and check out.
  • It would have been useful to have a little more structure to the activity than just going through the worksheet but on the sidewalk. Perhaps assigning roles, for example, scribe, coordinator, calculator.
  • The sun was hot meaning that students became lethargic towards the end of the lesson. Choose a shaded spot if possible.

Spaghetti Trigonometry – Extension

This is a direction I found myself going when some students had finished this activity.

– What would the graph of y = sin (2x) look like?
– y = 2 sin (x)?
– y = sin (x+2)
– y = sin (x) + 2?


What does this tell you about?
– f(ax)
– a.f(x)
– f(x+a)
– f(x) + a


Next lesson we are going to bring together everything we know about linear, polynomial and trigonometric functions and make some amazing graphical patterns using I hope to post the best ones here.

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?


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.