Partitioning & additionGrades 4–6

3 + 3 + 2

Eight eighth-notes in a bar. Cut them into groups and every grouping still adds to 8 — but each one feels completely different.

ArithmeticCombinatorics

A bar of 4/4 holds eight eighth-notes. Chop those eight into groups and you get a rhythm. Chop them 3 + 3 + 2 and you get the tresillo — the pattern behind half the music you already like.

tap a box to start a new group there

Tha

konnakol — the spoken counting NSM uses. Say it out loud with the drum.

it always adds to eight

3 + 3 + 2=8

3 groups, 8 eighth-notes, still exactly one bar of 4/4. Move the group starts around all you like — the total never budges, but the groove changes completely.

tempo96 BPM

Tick all eight underneath, so you can hear the grid the groups sit on

try another way of cutting up eight

From the NSM syllabus

“Tresillo using 8th Notes”
“Counting Subdivisions using Grid & Konnakol”
“Rhythmic Devices — Divide by 2”

Same total, different feel

3 + 3 + 2 = 8. So does 4 + 4, and 2 + 2 + 2 + 2. The arithmetic is identical every time. What changes is where the strong hits land — and that is the whole difference between a march and a dance.

Why 3 + 3 + 2 tugs

The beats are counted in twos: 1 & 2 & 3 & 4 &. Groups of three cut straight across that, so the second hit lands on an “and” instead of a number. Threes do not fit neatly into twos — that mismatch is the tug you feel.

Say it, then play it

Konnakol gives every group its own word: tha ki ta for three, tha kafor two. Say “tha ki ta · tha ki ta · tha ka” and you have said the tresillo out loud before you played a note.

For the classroom

Learning goals

Partitioning a fixed total: every way of splitting 8 into groups still sums to 8, so changing the arrangement does not change the sum.
Reading a group structure as an addition sentence — 3 + 3 + 2 = 8 — and hearing the difference that the arrangement makes.
Noticing that groups of 3 do not fit evenly into a count of 8 (or of 2s), which is exactly why the pattern feels off-balance.

Try this

1.Clap the eight eighths steadily while one half of the class claps only on the group starts of 3 + 3 + 2. Swap. Then write the addition sentence on the board.
2.Ask for every arrangement of 8 that uses only 2s and 3s. There are several — 3 + 3 + 2, 3 + 2 + 3, 2 + 3 + 3, and 2 + 2 + 2 + 2 — and each one sounds different.
3.Say the konnakol before you play it: 'tha ki ta, tha ki ta, tha ka'. Then play it and ask which syllables landed on a numbered beat and which landed on an 'and'.