Carrying in arithmetic is an exchange of quantities. An exploding-dots machine makes the exchange visible: a full group of dots in one box becomes a single dot in the next box to the left.
Start on paper with three boxes. Use a rule that two dots in one box may be exchanged for one dot in the box immediately to its left. Put five dots in the right-hand box and exchange groups until no box contains two dots.
Explain what remains
The result is one dot in the four-unit box, none in the two-unit box and one in the one-unit box. Written in base two, that is 101. The quantity remains five; the representation has changed.
Now change the exchange rule to ten dots for one. Place fourteen dots in the right-hand box. Exchanging a group of ten leaves one ten and four ones, represented by 14 in the familiar decimal system.
Ask the learner to predict an outcome before making the exchange. What happens with seven dots under the two-for-one rule? How can they show that the total value remains unchanged? Draw the groups if the symbols start to lose their meaning.
Use the Exploding Dots app to explore the same ideas with another base. Predict, make the exchanges and explain the result. Then try a fresh example away from the screen.
The useful evidence is the explanation: what each box represents, why the exchange is valid and how to recover the total quantity. An animation can help make the pattern visible, but the learner needs to account for the movement.