I explain that today we will be looking at a different time signature - 3/4 time instead of 4/4 time. In 3/4 time, there are 3 beats in each measure, and a quarter note is still considered one beat. That's where they get the "3/4" from. For our mathematical purposes, we will recognize that 3 quarter notes equal one whole measure, so we will represent the quarter notes as 1/3. This is something it's much better to show than say, but I'm wanted to make you aware of this possible misconception at the outset!
Here are some additional thoughts on how to teach the math part of three-fourths time.
I introduce the "thirds" first, represented in 3/4 time by quarter notes. Here is the example I use of three-four time with quarter notes, which we will be writing as 1/3 + 1/3 + 1/3 = 3/3. After we label it, either together on the whiteboard (I project this) and/or on their individual papers (here is the student page for thirds and sixths).
Next, I play the same piece of music with one change, each quarter note (1/3 of the measure) has been converted into 2 eighth notes (1/6 + 1/6 ). So now we are working with sixth notes in 3/4 time.
Finally, I put thirds and sixths together in the same several measures as a really clear example of how they relate to one another mathematically and musically.
My students were enjoying this, so we went through one more example of how thirds and sixths, represented in 3/4 time by quarter and eighth notes, can be combined.
I again have students get the mini-laptops and work with a partner as we create a several measure musical score in 3/4 time. It's important to spend this time building a solid understanding of the fractions in this very different context, as well as "teaching the tech", before sending them off to create their own musical masterpieces!
Here is a note to you on how to change the time signature.
As I did yesterday, I again ask the students to leave their musical scores up on the screen and I check them using a checklist, noting who was able to complete the task successfully, who is "almost there," and who was having some technical or mathematical difficulties.
I ask students to think in their head of two examples of equivalent sets of notes or fractions.
These are some answers I heard from my students: