We integrated the ”Make It Sound” international Smart Hands workshop in the maths lesson *Ratios, proportions and mean(arithmetic and harmonic mean) *for the 9^{th} graders from Colegiul National Emil Racovita.

The lesson started with a structured discussion around the following questions:

- Do you know any string instruments? Do you know anything about the arithmetic or geometric mean of 2 positive numbers? Or the harmonic mean of 2 positive numbers? What about the octave, quint or quart?
- Is there any connection between the 3 means of 2 positive numbers (or just one of them maybe) and the octave, quint and quart? If so, which is the connection?

Then, the students were asked to make monocords (the process has been tested in the Make It Sound workshop) with different string lengths: 90cm string length, 45cm string length, 60cm string length, 67.5cm string length and to compare the sounds produced by the 4 monocords.

In groups of 4 (each student in the group had a different string length for their monochord) the students worked together to investigate the connection between the sound and the length of the string and to identify different properties of musical proportion. During the group work activity, the students solved the tasks related to the musical proportion, the relationship among the musical proportion and the proportion made with 2 positive numbers, the arithmetic mean and their harmonic mean, harmony of sounds, the frequency of the sounds produced by the 4 monocords (they used FizzIQ app to measure the frequencies) and the relationship between the musical notes and the string frequency.

The lesson ended with the students’ reflection on their work.

### What do students say about this lesson?

“*Throughout the lesson I learnt about the way in which the length of a cord could influence the frequency of the sounds it emits. I also received information about a thing I had known nothing up to that moment: the musical proportion; I find it very interesting that maths plays an important role even in music. I was amazed by the way in which we can find the length of the cord that emits an octave, a fifth or a quartet of a sound by using a basic calculation. I also enjoyed working in a team, which allowed us each to focus on an exercise and take the necessary time to think about it. Even though we found some of the questions difficult, we managed to find the correct answer and solve the problems using both mathematical knowledge and logic. I believe everybody enjoyed their time during class and found the activity less stressful. All in all, it was an interesting lesson that also had a practical part*.” (Amelia P.)

“*During this lesson I found out what Pythagoras had discovered by using the monocord and I acknowledged the link between maths and music. Based on ratios and the 4 intervals, Pythagoras could come up with the diatonic scale whose notes were called C, D, E, F, G, A, B and C, and basically created the foundation of musical theory using number sequence and ratios.*“

*Using our four-string monocords, our team found out that when the strings vibrate together, they produce 4 sounds, the shorter strings having the highest pitch( an octave) and the longer string having the lowest pitch.*

*I really enjoyed this interesting experiment and didn’t have any difficulties, except for the type of cord I should use to get a good sound. This type of lessons makes me understand Maths better. “*(Ana S.)