## Making Simple Ratios (Activity for Teachers)

Using the Impromptu app, you can make this 2:1 beat hierarchy and listen to it. Listen to Figure 2 for an example.

Figure 2

How does this work? At the right in Figure 2 are drumblocks. Drumblocks play percussion sounds when they are dragged into the Playroom. The numbers on drumblocks are based on two general principles:

1. Larger numbers represent longer durations; smaller numbers represent shorter durations. The smaller the number on a drumblock, the faster percussion events will follow one another.

2. The numbers representing durations are proportional to one another: e.g., a series of 3-blocks makes a beat that goes twice as fast as a series of 6-blocks; a series of 12-blocks makes a beat that goes twice as slow as a 6-block series.Text

### Do it yourself…

1. With Impromptu open on your computer, select Drummer in the PlayRooms menu. (Click here to watch a brief tutorial on the Drummer Playroom)

2. Drag drum blocks into each voice (level) as shown in Figure 2.

3. The boxes on the left indicate how many times a drum block should repeat.

4. Touch the space bar, listen, and watch the graphics below.

Figure 3

Impromptu uses two kinds of graphics to represent rhythms—rhythm bars and rhythm roll.  Both rhythm bars and rhythm roll represent time relations through analogous spatial relations. Looking now at the picture of the Impromptu screen in Figure 4, you see that in rhythm bars graphics, spaces between lines match the time between events (claps).

Figure 4

For instance, the unequally spaced vertical lines in the top row of the graphics window represent the varied durations of the rhythm of the tune, Hot Cross Buns; the equal spaces between lines in the other three rows of the graphics represent the equal duration beats played by the different percussion instruments.  Events that take up more time (go slower), also take up proportionately more space.

Once made, we can ask questions to find the math in how this grid works.  For instance:  Why do the three percussion instruments, each playing a steady beat, produce 2:1 proportional time relations among them?  Consider the respective durations of the drum blocks as you move from one level to the next:

8:4 = 2:1; 4:2 = 2:1

Listening to the percussion accompaniment, it sounds as if all the instruments end up together.  However, it doesn’t look that way in the rhythm bar graphics.  What if we try rhythm roll graphics as in Figure 5?  (Click here to watch a short tutorial on changing the graphic views for the drummer playroom)

Figure 5

Now you can see the instruments converging at the end.  What makes the difference?  In rhythm roll graphics the time between events is “filled in”—as if you could see time passing. The rhythm bars graphics show only the very brief clap sound not the gap between.

What about the repeats in the boxes on the left of each line?   Like the beat values, the repeats also have 2:1 proportional relations (8, 16, 32).  But, as the children say, it’s “upside down” compared to the ordering of the beat values (8, 4, 2). That is, the beat values get proportionally smaller (beats go faster), while the corresponding number of repeats get proportionally larger.  But why?

One way to think about it is that the beat values tell you “how much” time for each event, while the repeat values tell you “how many” events there are.

Beat Values      Repeats

8                      8

4                    16

2                   32

Anna, a student in 5th grade, made it quite clear when she said:

“…like 3 is twice as fast as 6, so the ‘repeat’ has to be twice as much, too.”

Figure 6

Sam, starting out with beats of 4 and 2, said it a different way:

“It works because the 2 is half as big, so it gets twice as many repeats as the 4. I mean the ‘twice as much’ is the same but it’s in reverse: 4 is to 2 like 6 is to 12 except upside down.”