Disco Balls

Prime factors made visible: a product becomes circles inside circles, so multiplication reads as structure.

Grades: 2–6
Subjects: Multiplying & Dividing
Kind: Puzzle
Cost: Free, no account, no ads

What it is

The tray holds 2, 3, 5 and 7 and nothing else. Lay them in a row and the row draws itself: a recipe of 2 × 3 is one big circle holding two circles, each of those holding three, and the innermost circles painted solid black. Count the black dots and you have the product.

What it gives you to practise

Prime factorisation, met as a picture before it is met as a procedure. There is no tile for 6 because 6 is already 2 × 3, and no tile for 12 because 12 is 2 × 2 × 3 — so an order for 72 dots decides your tiles for you, 2, 2, 2, 3, 3, and no other bag will do. What is left to choose is the order, and the order changes the picture completely while leaving the count exactly where it was. That is commutativity with something at stake: two pictures that look nothing alike, the same ten dots in both.

In a classroom

The switch that turns the nest inside out into a branching tree is the one worth showing on a screen: the same recipe, the same dots, drawn twice in two ways, with nothing added and nothing taken away.

The rules

Build a multiplication out of prime numbers and watch it draw itself — as circles inside circles, or as a branching tree.

How the picture works

Your recipe is a row of numbers, read from the outside in. A recipe of 2 × 3 means: the big circle holds 2 circles, and each of those holds 3 circles.
The innermost circles are painted solid black. Those are the dots.
Count the dots and you have the answer to the multiplication. 2 × 3 makes 6 dots. 2 × 2 × 3 makes 12.
Every circle packs its circles as tightly as it can, and each new ring is turned relative to the circle holding it — which is why the picture keeps growing new shapes as your recipe gets longer.
That turn is yours to set: spin the picture and the outer ring follows your finger while everything deeper winds faster, like gears.

Building a recipe

The line above the picture says how many dots this order is asking for, with the number written large. It lights up when your recipe has grown exactly that many.
Tap a number in the tray to add it to the end of your recipe.
Drag a number from the tray into the recipe row to put it exactly where you want it. A space opens up as you aim, the size of the number and with its × signs already written, so the sentence you are looking at is the one you will get.
Drag a number onto a circle in the picture instead, and that circle comes to hold that many. Drop a 3 on the big outer circle and the whole picture becomes 3 circles; drop it on a black dot and every dot becomes 3 dots. Whichever circle you point at lights up, and the space in the recipe opens at the matching place in the multiplication — that is where that circle lives in the sentence.
Drag a number that is already in your recipe to move it somewhere else — to another place in the row, or straight onto a circle. Drag it right out of the row to throw it away — or just tap it, and out it goes.
Tap two numbers in your recipe to swap them.
Press Done when the order is filled, or Clear to clear the recipe.

The three switches above the picture

Tree turns the nest inside out: every circle flies out of the one holding it, the circles become branches, and your dots end up at the tips. Nest packs them back. Nothing is added or taken away — count the dots either way and you get the same answer.
Colour gives every number its own colour, so you can see which tile built which ring, or which branch. Press it again for greys.
Twist ↻ sets the whole picture turning like a wheel, slowly, until you press Stop. Drag the picture any time you like — even while it is turning — and flick it to send it spinning. In the nest each generation winds faster than the one outside it, like gears; the tree turns all in one piece.

Reading the shades

The circles get darker the deeper they go, so you can always tell which ring is inside which.
The innermost circles are the darkest — that is how you spot the dots to count, however deep the nest.
On the tree the branches darken the same way as they go out, and only the tips are dots: nothing else on the board is a circle, so there is nothing to miscount.
With colours turned on, the shade still tells you the depth and the colour tells you the number: a pale blue ring and a dark blue ring were both made by a 2.

The big secret

Change the order of your recipe and the picture changes completely — but the number of dots stays exactly the same. 2 × 5 and 5 × 2 both make 10 dots, in two very different pictures.
The tray only ever holds 2, 3, 5 and 7 — the small prime numbers. A prime cannot be split into a smaller multiplication, so these are the seeds everything else is built from.
There is no tile for 6, because 6 is already 2 × 3. There is no tile for 12, because 12 is 2 × 2 × 3.
So an order for 72 dots decides your tiles for you — 2, 2, 2, 3, 3, and no other bag will do. What is yours to choose is the order, and that means the picture.

The packs

First Blooms — twos and threes, shallow nests, small counts.
Prime Bouquets — fives arrive, and the same count turns up in two very different pictures.
Deep Nests — sevens, and nests deep enough to shrink the dots to seeds.
Wild Gardens — orders that describe a kind of number instead of naming one: a count between 50 and 60 with exactly 4 tiles, a square number of dots, one of every tile. Several answers usually work.

Play Disco Balls